The Two-Layer Hierarchical Distribution Model of Zeros of Riemann’S Zeta Function Along the Critical Line

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Article The Two-Layer Hierarchical Distribution Model of Zeros of Riemann’s Zeta Function along the Critical Line

Michel Riguidel

Department of Computer Science and Networks, Télécom ParisTech, 75015 Paris, France; [email protected]; Tel.: +33-67687-5199

Abstract: This article numerically analyzes the distribution of the zeros of Riemann’s zeta function along the critical line (CL). The zeros are distributed according to a hierarchical two-layered model, one deterministic, the other stochastic. Following a complex plane anamorphosis involving the Lambert function, the distribution of zeros along the transformed CL follows the realization of a stochastic process of regularly spaced independent Gaussian random variables, each linked to a zero. The value of the standard deviation allows the possible overlapping of adjacent realizations of the random variables, over a narrow conﬁdence interval. The hierarchical model splits the ζ function into sequential equivalence classes, with the range of probability densities of realizations coinciding with the spectrum of behavioral styles of the classes. The model aims to express, on the CL, the coordinates of the alternating cancellations of the real and imaginary parts of the ζ function, to dissect the formula for the number of zeros below a threshold, to estimate the statistical laws of two consecutive zeros, of function maxima and moments. This also helps explain the absence of multiple roots.

Keywords: Riemann’s zeta function; non-trivial zeros; Lambert’s function; Laplace–Gauss law; gaps; multiple roots

Citation: Riguidel, M. The Two-Layer Hierarchical Distribution 1. Introduction Model of Zeros of Riemann’s Zeta This article presents a distribution model of the non-trivial zeros of the zeta function Function along the Critical Line. on the critical line (CL). The Riemann Hypothesis (RH) [1] places them exclusively on this Information 2021 12 , , 22. https:// line, but this article focuses on the CL zeros, regardless of the RH. doi.org/10.3390/info12010022 We ﬁrst explored the global behavior of the ζ function on the CL, and the role of the Lambert function [2] on the folds of the ζ function in the critical strip (CS). A close Received: 11 December 2020 relationship exists, on the CL, between Riemann’s ζ function and Lambert’s W function, Accepted: 4 January 2021 Published: 8 January 2021 after an anamorphosis of the ordinate of the complex plane. This article also shows that, on the CL, there is a relationship between both the real

Publisher’s Note: MDPI stays neutral and imaginary components of the ζ function. This ratio, only valid on the CL, takes into with regard to jurisdictional claims in account an angle tied to the fractional part of the anamorphosed ordinate, which makes it published maps and institutional afﬁl- possible to anchor the common zeros of these two components in an equivalence relation, iations. where the classes are intervals of unit length. Thus, both ζ surfaces, alternatingly zero, with period one on the anamorphosed ordinate of the complex plane, locally generate an additional common zero which is the nth non-trivial zero. The article then investigates, on the anamorphosed axis, the local “random” character of the nth non-trivial zero, in the form of the realization of a Gaussian random variable (RV) Copyright: © 2021 by the author. n σ W( n) Licensee MDPI, Basel, Switzerland. with a mean and standard deviation en increasing very slowly with ln , of order 1 3 < 7 This article is an open access article / for n 10 , all these n RVs being pairwise independent. The magnitude of the distributed under the terms and standard deviation leads to the possible permutation of some realizations of adjacent RVs, conditions of the Creative Commons justifying the uncertainty in the formula for the number of zeros below a given threshold, Attribution (CC BY) license (https:// and explaining the corresponding difﬁculty for algorithms to compute these zeros. creativecommons.org/licenses/by/ Finally, this article presents a taxonomy of the local behaviors of the ζ function in the 4.0/). CS, according to the probability density of the RV relevant to the nth zero. The model still

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Information 2021, 12, 22 Finally, this article presents a taxonomy of the local behaviors of the function in 2 of 35 the CS, according to the probability density of the RV relevant to the th zero. The model still allows a number of derivations which reaffirm some classical formulas of the Rie- mann function. This also helps explain the absence of multiple roots. allows a number of derivations which reaffirm some classical formulas of the Riemann ζ By consulting the abundant scientific literature on the zeros of the Riemann func- function. This also helps explain the absence of multiple roots. tion, including certain pioneering articles [3–8], one can distinguish three broad families By consulting the abundant scientific literature on the zeros of the Riemann ζ function, of publications, with the first two having the underlying ambition to provide proof of the including certain pioneering articles [3–8], one can distinguish three broad families of RH. publications, with the first two having the underlying ambition to provide proof of the RH. The first family consists in circumscribing the perimeter of the region where the zeros The first family consists in circumscribing the perimeter of the region where the zeros are located. As early as 1896, Hadamard [3] and La Vallée Poussin [4] restricted the “fertile are located. As early as 1896, Hadamard [3] and La Vallée Poussin [4] restricted the “fertile 0<<1 region” ofregion” zeros to of the zeros CS to the CS (0 <. Then,x < 1 during). Then, the during twentieth the twentieth century, the century, region the was region was reduced inreduced two ways. in two On ways. the one On hand, the one by hand,further by limiting further the limiting perimeter, the perimeter, and on the and other on the other hand by handincreasing by increasing the percentage the percentage of potential of potentialzeros on zerosor around on or the around CL [9,10], the CL in [9an,10 ], in an increasinglyincreasingly restricted restricted perimeter. perimeter. The secondThe family second consists family in consists analyzing in analyzing only CL zeros only CL[11,12], zeros by [11 estimating,12], by estimating the num- the number of zerosber ofless zeros than less a than value a y ofvalue the ofordinate the ordinate of [13], of s [by13 ],calculating by calculating the number the number of of zero zero crossingscrossings on the on theCL, CL,by studying by studying the the adja adjacentcent zeros, zeros, or orby by estimating estimating other other charac- characteristics teristics ofof the ζ function, function, always always with with the the aim aim of apprehending of apprehending theRH, the accordingRH, according to various to points various pointsof view. of view. The thirdThe family, third more family, concretely, more concretely, consists in consists effectively in effectively calculating calculating the CL zeros the CLwith zeros with a numericala numerical algorithm algorithm [14,15]. Currently, [14,15]. Currently, the exercise, the exercise,which thrived which between thrived between1985 and 1985 and 2005, has2005, seen hasa loss seen of amomentum, loss of momentum, as the nu asmber the number of known of knownzeros is zeros substantial. is substantial. Extend- Extending ing thesethese calculations calculations has become has become superfluous superfluous with withsilicon silicon computers, computers, although although searches searches for for zeros zerosbeyond beyond aboutabout > 10n > 10 over100 anover interval an interval of the of order theorder of Δ of =∆ 10n =, 10would10, would pro- provide vide interestinginteresting additional additional knowledge knowledge on the on the function.ζ function. The aim ofThe this aim article of this belongs article to belongs the second to the family, second by family, taking by ad takingvantage advantage of numerical of numerical results ofresults the third of thefamily. third Indeed, family. the Indeed, goal here the goal is not here to isinvestigate not to investigate questions questions about the about the RH, nor RH,to determine nor to determine an algorithm an algorithm to effectively to effectively calculate calculatezeros. The zeros. goal Theis mainly goal is to mainly to define, ondefine, the CL, on a the distribution CL, a distribution model with model two with layers, two one layers, deterministic one deterministic and the other and the other stochastic,stochastic, to specify to the specify local thecharacteristics local characteristics of the function of the ζ functionaround the around zeros the and zeros to and to discuss thediscuss evolution the evolution of local behavior of local behaviorof rare events of rare with events increasing with increasing . n. Despite itsDespite age, Edward its age, Charles Edward Titchmarsh’s Charles Titchmarsh’s book [16] book is the [16 most] is thecomprehensive most comprehensive book on book function on ζ function theory. theory. It covers It covers almost almost everything everything anyone anyone would would want want to know to know about about thethe ζfunction.function. Aleksandar Aleksandar Ivi Ivić’sbookć’s book ( (AАлександарлексaндap Ивић))[ [17]17] offersoffers anan easier easier approach approachfor for anyone anyone who who wants wants to to understand understand all all of of the the questions questions about about the theζ function, function, including includingthe the theory theory but but also also its its applications. applications. After thisAfter introduction, this introduction, the notations, the notations, methodology, methodology, data, computer data, computer science and science and mathematicalmathematical tools are toolspresented are presented in Section in 2. Section Section2. 3 Section presents3 presents the results. the results.A discussion A discussion of the assumptionsof the assumptions and results and is resultsexpressed is expressed in Section in 4, Section before4 the, before conclusion. the conclusion. The whole The whole article is basedarticle on is basednumerical on numerical calculations calculations performed performed by computer by computer programs programs in the Python in the Python language,language, designed designed and implemented and implemented by the au bythor. the Some author. calculations Some calculations are supported are supported by by power series,power which series, were which validated were validated by computer. by computer. 2. Materials and Methods 2. Materials and Methods 2.1. Mathematical Notations 2.1. Mathematical Notations Where possible, classical mathematical notations have been selected (Table1): to lighten Wherethe possible, writing, someclassical original mathematical notations arenota defined.tions have been selected (Table 1): to lighten the writing, some original notations are defined. 2.2. Numerical and Graphical Observation as a Tool for Reflection It is fruitful to numerically represent and visualize the catalog of formulas, well known to all, to verify some exact formulas with infinite sums and to validate approximations or inequalities, demonstrated mathematically, to testify to their robustness, speed of convergence and power of truth. Graphical visualization is a powerful tool for analysis and reflection, which amply guides intuition to take paths that are difficult, if not impossible, to take with the traditional manipulation of mathematical formulas. However, the ζ function calculations become complicated, even impractical, on a computer, when n takes

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Information 2021, 12, 22 Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses3 of 35high values more than the Logarithm function.

2.5.5. Hadamard’s Product Formula Information 2021, 12, x FOR PEER REVIEW 3 of 35 Information 2021, 12, x FOR PEER REVIEW 100 3 of 35 large values (n > 10 ).Hadamard’s The window product of practicable formula computer contains calculationsin its expression, ignores on large the one hand, the non- numbers. trivial zeros and on the other hand, the functional Riemann equation. It therefore has a remarkable strength to characterize the zeta function and its deep relationship with neigh- Table 1. Notations. Table 1. Notations. borhoodsTable 1. of Notations. zeros. We know the formula: DefinitionDeﬁnition Definition ′() Notation 1 Γ′(1+½) Notation 1 1 Absolute value = ln2−½−1− −½∀ ∊ ℝ: || =+ > 0; − + < 0 Absolute valuevalue (∀∀)a ∊∈ℝR: :|||a|== a i f a > 0; 0; − −1−a i f a < <0Γ( 01+½ ) − () iθ i arg ∀(z) ∈() ℂ: = + = = || ; ∀∀z ∈∈ ℂ: : z == x ++ iy == re ==|z||e| ; ; (½) C Moreover, we know that:p ∑22 =1+ |⁄| 2=−ln2−½ln;|+| = + ; ,= , , (1+½ ∈ ℝ ) ||z|| = ||+x + iy||== x ++y ; x ;, ,y, ,r, ,θ ∈ ∈Rℝ (½) Complex number = (⁄) 2 > 0; ComplexComplex numbernumber The formulaθ == becomes: arctan(y/ ⁄)x) 2 modulo 2π i f x > 0; > 0; θ = arctan(y/x) modulo π + π i f x < =0; (⁄) + < 0; = (⁄) + < 0; = ′=() > = − =0: =< 1⁄ 2 > 0; 1 = −⁄ 2 < 0 =0: =x 0 : θ π⁄/2 2= ln i f y √ >0; −½ θ = −(π1+½/2⁄ 2i f y ) − <0 0 + () In ℝ, the integer part−1 and the fractional − part of a real number : Integer andIn ℝfractionalIn, Rthe, the integer integer parts part part and and the the fractionalfractional part part of of a reala real number numbera: : IntegerInteger and fractionalfractional parts parts ∀ ∊ ℝ: =[] + {}; [] ∊ℕ,0≤{} <1 Using a∀∀ CLa ∈ ∊ pointRℝ:: a = [ aof]]++ ℒ{{a }and};; [a[ ]the∈] ∊ℕ,0≤N zeros, 0 ≤ {⊂ℨa{}}<<11 on the CL ℒ⊂, which assumes the Complex number in Informationthe CS 2021, 12 In, x FORℂ, =+ =ℜ PEER REVIEW () + ℑ();∈ℂ;,∊ ℝ; its conjugate ̅ =− 18 of 35 ComplexComplex number in in the the CS CS In ℂ In,RH C=+ =ℜ, s to= bex + true,i y = theR(( sinfinite)) + ℑ+ i ( ssum));;∈ℂ;,∊s ∈ isC written:; x, y ∈ R ℝ∑;; its its conjugate conjugate=−s =∑ x̅ =− − i y In ℂ, ̂ =1−=1−− ;,∊ ℝ; 1−We finallyInIn obtain,ℂC, , sˆ̂ ==1−=1−− ;,∊ taking1 − s = into1 − accox − iunt y; x ,they ∈ pairsR;ℝ ;of zeros on the CL (>0 and <0): Complex number 1− associatedComplex with number associated with ̅ Complex number 1 − s associated with s its conjugate sˇ̌ ==1−1−s==sˆ =̂̅ =1−+ 1 − xits+ conjugatei y. ̌ =1−=̂ =1−+ . its conjugate . ′sˆ (symmetric) of s with respect toalternation (1̂ , 0).symmetric1 that ofgenerates 1 with arespect non-trivial1 to (½, zero 0). in a restricted interval. On the CL, the calcu- ̂ symmetric= ln√ of −½ with( respect1+½ )to2+ (½,− 0). + Critical strip (CS)S =(R)( ) + ( ) = In +the CS ≤lations: =ℜ≤̅ are(> extremely) + ℑ6=(−) = + ; 0≤≤1,>0; ≠1simplified: + . CriticalCritical stripstrip (CS) (CS) In Inthe the CS CS : :=ℜs (s ) + ℑi (s) = + ; 0≤≤1,>0; ≠1x i y; 0 x 1, y 0; s 1. . ℒ∶==½1 =½+; >0 Critical line (CL) Critical line (CL) ℒ∶==½L = = =1 =½+; (=½+; >0=) 1+2+ On the(> )CL= () (̂) =: ()()̅ =()().⇒ |()| =1⇒() = Critical line (CL) OnOn the the CL CL : x 2 : : 2 i ; 0..− CS, except the CL \ ℒ∶=0≤≤1;≠½; ≠1 CS, except the CL \ ℒ∶=0≤≤1;≠½; ≠11 CS, except the CL S\L := 0 ≤ x ≤ 1; x 6= 2 ; s 6= 1 This formula converges slowly on( the) = imaginary∑ . Incomponent, the CS,( the⁄) because function of( the⁄) is termsdivergent. in () = ∑ . Γ ̂ 2 Γ 2 Zeta function:( ) = ∞ − s In the CS, the function is divergent.() = = ; ()() =1 Zeta function: . ζ s ∑1 n . In the CS, the ζ function is divergent. () Zeta function: ζ One considers in this CS, the analyticalOne considers continuation in this CS,() Γ.the (2 ⁄)analytical Γcontinuation(2⁄) . One considers in this CS, the analytical continuation η(s). (⅝) Êta DirichletWhen function: is very large, we can do the approximation:() = ( )(≈1−2⅝ )≈. . Êta Dirichlet function: () =()( 1−2−s+1 ). Êta Dirichlet function: η η(s) = ζ(s) 1 − 2 . By multiplying by () if is not a zero: () = ()()() =()|()| Derivative of the function () =−∑ log()⁄ Derivative of the function The sum can then ( be) =− broken∑ downlog( into)⁄ two sums if ⋘ : Derivative of the ζ function ζ0(s) = − ∞ log(n)/ns ∑n=1 ()( )(∑) ( ) ℑ() () = −log ⁄ k-order derivatives of the function () = ∑1 −log|()|(≠0 ⇒)k 1( ) = =1 ⇒ () = ±|()| ⇒ =tan k-order derivatives of the function ζ(k)(s) = ∑∞ (− log(n)) /ns |()| ( ) 2 k-order derivatives of the ζ function n=1In the= CS, one uses the + analytical continuation of these derivativesℜ In theIn the CS, CS, one one uses uses the the analytical analytical − continuationcontinuation of of these− these derivatives derivatives − () =()(̂) =()(1−) () =()(̂) =()(1−) ( ) =−( )|( )| Functional equationWe of canthe then function ζchoose(s) = ξ (s) ζto(sˆ) obtain= ξ(s) ζa( 1good− s) Ifapproximation, is zero, Hospital’s but the ruleconvergence gives: Γ( iŝ⁄) 2al- Functional equation of the function ( ) Γ(̂⁄)2 ( ⁄) ( ) ½ Functional equation of the ζ function s s−1 s−=2½1 Γ(sˆ/2 ) sin 2 Γ 1− = ways slow.(ξ ()s=2) = 2π sinsin((πs⁄)/2 2 )Γ((11−− s))==π 2 Γ(2⁄) Γ(Γs/2(2)⁄)′( ) ℜ () |()| =0 ⇒() =− = ⇒ () = ±̅ ()⁄ ⇒− =tan If was always very small compared (to) ̅ = ,( which)(̂) =is never()( the)̅ = case() since(); (())==( ())(ˆ̂)==(())()̅ == ()(); |′()| ℑ () 2 Functionalsweeps equation the on ζentire the CL rangeξ ζ of ordinates,ξ ζ we coulξ d haveζ ; an estimate ofΓ the(2⁄) infinite sum: FunctionalFunctional equation on on the the CL CL Γ(2⁄) ( ) ( ) Γ( /2) = ⁄ ξ() == πi The non-trivial zerosΓ( 2are⁄) at the breaking point of the two functions + and 1 1ΓΓ((2/2⁄)) ln − 1 ⋘ ⇒ ≈ −⁄ . t=⁄(2) = ⁄ 2 t=⁄(24) = ⁄ 2 t t= y/̃ =ln(2π) =(y⁄)/2π=⁄(2Using) ln( the⁄)(2 power) . The series anamorphosis expansion of produces Γ(2⁄), awe stronger serially expand () to find : ̃ Anamorphosis:=ln(⁄)=⁄ (2̃ ) ln(⁄)(2)1 . The1 anamorphosis + 2 + 2 ln 2 produces + () −24ln a stronger Anamorphosis: ̃ et = t ln(t/e) = y/(2π) ln(y/(2πe)). The anamorphosiselongation produces of a the stronger axis as increases. Anamorphosis: t elongation of= the axis as| ( )|increases. ( ) () ⁄ { ̃} ⁄( ) e elongation of4 the y axis as yincreases. =1⇒6 = = ; = 4 −2 ; = −1 24 ; We define by the adjective “digressive”We define a series by the that adjective exhibits “digressive” a beginning a simi-series that exhibits a beginning simi- We deﬁne by the adjective 1 “digressive”1 lar to a a series known3i that series, exhibits9 but a beginning which139i varies similar 475noticeably to with thei subsequent1 terms.73i 131 lar to a known series, but which varies noticeably with( ) the subsequent terms. a known series, but which≈ 1 varies + 2=1− noticeably + 2 ln 2 with− + ′′ the2 subsequent−ln+ terms.+ +⋯1+ − − + +⋯ 24 For example,8 128the cosine 3072 series is written:32768 cos = 1 − 3+ 18− +⋯+1620 9720 For example, the Forcosine example, series the is cosinewritten: series cos is written: = 1 − + − +⋯+ ! ! ! 2 4 6 2n ! ! ! Digressive series Digressive seriesx x x n x 2n+1 Digressive series cos x = 1 − + − + ··· +((−1−1)) +ℴ+ (x ).. WeWe will will then then note, note, in order to simplify the writing, a (−1) +ℴ(2! 4! ). We6! will then note,((2n) !in iorder to simplify1 (the6×7×24−5 writing, a )i 7×24 −5 19 × 31 × 8707i 317 × 531103 ()! in order to simplify the writing, a digressive cosine: = =1− − − − − − a x2 a x4 a xdigressive6 24 cosine:n2!a x( 224n ()) =1−5×3! (24+ ) − 5×4!+⋯+(24(−1) ) 7×5!+(24) 35 × 6! (24) digressivecos˘ (x cosine:) = 1 − 2 +()4 =1−− 6 + ···+ + (−−1) 2n +⋯++ (x−12n+)1 . ! + ! ! ()! 2! 4! 6! ! ! 13 ×!( 5243712823i2n)! (31) ×! 661 × 28660561 27 × 8197117457520809i ℴ() − − . − Complex zeros Z in the CS S zn = xn ℴ+(i yn; ζ()z. n) = 05×7!(24) 5×8!(24) 25 × 11 × 9! (24) Complex zeros ℨ in the CS = + ; () =0 Complex zeros ℨ in the CS = 1 + ; (81) =0 × 5099 × 4517050132181 n = + i n; ζ( n) = 0 2 − =½+ ; (+⋯) =0 Zeros Z ⊂ Z on the CL L ⊂ S The zeros are=½+ ordered in; pairs,()s=0and11×5×10! 1 − s. (24) ⊂ℨ ℒ⊂ The zeros are ordered in pairs, and 1−. Zeros ⊂ℨ on the CL ℒ⊂ Zeros on the TheCL Calculationszeros are ordered are carried in pairs, out with andy > 01−. CalculationsWe can therefore are carried extract out withthe rotation>0 ==−1⁄(24) from this expansion: Calculations are carried out with >0 = + () Lambert function The main branch is defined by: () = ⇔= ;≥−1⁄ . Lambert function The main branch is defined by: () = ⇔= ;≥−1⁄ . Γ() = Gamma function ( ) = + ()≈=−1⁄(24) Gamma function Γ = Euler’s constant Euler–Mascheroni constant: = lim− ln + ∑ ≈ 0.5772156649 Euler’s constant Euler–Mascheroni constant: = lim− ln +∑ is a three-order ≈ 0.5772156649 polynomial→ in = −⁄(4! ): → () () () Taylor series ( ) () = ∑ (−) Taylor series () = ∑24 (−) 17807 29 ×! 10079 43 × 11009087 13 131 × 23027 = ! 7 − 7 + − + − Bernoulli numbers 5 Coefficients 7×2! of the power7×5×3! series of = ∑7×5×4!ℬ 7×5! Bernoulli numbers Coefficients of the power series of = ∑ ℬ ! 3458158927391587 ! 7 × 47 × 2095343 × 148000093 + − +⋯ 3×5×7×9×11×6! 3×5×7×9×11×7!

168 = 1 + 5 4 36 + 31 × 12 + (34 × 239 + 2 × 1643717 + 53 × 66797) 35 5 1 + (68 × 61 × 107923 × 965611 + 23 × 4881469 × 16227313) +⋯ 4511 We can deduce an approximate value of () on the CL: 168 ≈ (1+); = −1⁄(24) 5

168 168 () = ≈ + (1+)≈() 1 + ≈ ≈() 5 5

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Table 1. Cont.

Definition Notation Lambert function The main branch is defined by: W(z) = w ⇔ z = wew; z ≥ −1/e . R ∞ z−1 −t Gamma function Γ(z) = 0 t e dt Information 2021, 12, x FOR PEER REVIEW 4 of 35 Euler’s constant Euler–Mascheroni constant: γ = lim − ln n + ∑n 1 ≈ 0.5772156649 n→∞ k=1 k (n) Taylor series ∞ f (a) n f (z) = ∑n=0 n! (z − a) t ℬ= =1; ℬ∞ t=−½; >0:ℬn =0 ;ℬ = ;ℬ =− ;ℬ = ;ℬ = Coefficients of the power series of et−1 ∑n=0 n n! = = − 1 i f k > = = 1 = − 1 = 1 = 0 1; 1 2 ; 0 : 2k+1 0 ; 2 6 ; 4 − 30 ;;ℬ6 =42 ; ;ℬ8 =− ;ℬ = ;ℬ =− ;ℬ = ; Bernoulli numbers 1 5 691 7 3617 43,867 − ; 10 = ; 12 = − ; B14 = ; 16 = − ; 18 = ; ()! 30 66 2730 6 510 798ℬ = (−1)|ℬ |; ℬ ~(−1) n−1 n−1 2(2n)! () n = (−1) n ; n ∼ (−1) 2n 2 2 2 (2π) 2.2. Numerical and Graphical Observation as a Tool for Reflection It is fruitful to numerically represent and visualize the catalog of formulas, well 2.3. The Gap between Observation and Demonstration known to all, to verify some exact formulas with infinite sums and to validate approxima- Numerical processing and observation allowtions for the or quashinginequalities, of a demonstrated formula, approxima- mathematically, to testify to their robustness, speed of tion or inequality, with a single representation, whenconvergence this representation and power does of truth. not conform Graphical visualization is a powerful tool for analysis to the postulate or original intuition. However, theand numerical reflection, calculation which amply and guides graphical intuition vi- to take paths that are difficult, if not impos- sualization on the first few million zeros do not demonstratesible, to take anything. with the We traditional know the manipula fallacy tion of mathematical formulas. However, the of the hasty general conclusions drawn from premature function examinations calculations of abecome phenomenon complicated ob- , even impractical, on a computer, when served over too small an interval, while the results change or reverse over a wider interval. takes large values ( > 10). The window of practicable computer calculations ignores The well-known example is the Skewes number on the difference π(x) − Li(x), where the large numbers. sign swaps for huge values (a current upper bound where this occurs is 1.39822 × 10316). For the ζ function, the profile of the real and imaginary surfaces has more and more tightly 2.3. The Gap between Observation and Demonstration packed folds, with increasingly strong amplitudes. The mixing of RVs and the rules for the appearance of zeros undergo very slow evolutions, soNumerical that the observationsprocessing and for observationn < 1020 allow for the quashing of a formula, approx- risk masking new landscapes of the ζ function forimation higher or values, inequality, beyond with10100 a single. A (perhaps representation, when this representation does not con- daring) extrapolation will nevertheless help to predictform to the the behavior postulate beyond or originalN = intuition.1020. However, the numerical calculation and graph- The methodology here consists in first presentingical visualization the graphs andon the statistics first few on themillion raw zeros do not demonstrate anything. We know variables. An analysis then allows these variablesthe to fallacy undergo of the geometric hasty general transformations, conclusions drawn from premature examinations of a phe- in order to obtain more abstract calculated variablesnomenon but with observed classic over statistics too small that canan interval, be while the results change or reverse over a manipulated, mathematically. wider interval. The well-known example is the Skewes number on the difference () − Observation is also a good complementary() tool,, precedingwhere the asign demonstration. swaps for huge This values is (a current upper bound where this occurs is what we will expose by presenting the graphical1.39822 evolution× 10 of). the For local the distribution function, the of profile of the real and imaginary surfaces has consecutive zeros and the evidence for the presencemore of and only more single tightly roots packed in the CS. folds, with increasingly strong amplitudes. The mixing of RVs and the rules for the appearance of zeros undergo very slow evolutions, so that the 2.4. Data and Computer Science Tools Used observations for <10 risk masking new landscapes of the function for higher val- This study was performed using the set D,ues, which beyond represents 10 the. A first(perhaps 6 million daring) zeros, extrapolation will nevertheless help to predict the as well as a few other sets of higher values. Onbehavior the interval beyondI := =100 < y <. 1020 , we observe, without exception, that the zeros are located onThe the methodology CL (x = 1/ 2here) according consists toin thefirst presenting the graphs and statistics on the raw two-layered model (Lambert function and stochasticvariables. Laplace–Gauss An analysis process)then allows described these variables to undergo geometric transformations, below. To obtain the list of zeros on the CL, onein can order consult to obtain several more websites abstract on thecalculated Inter- variables but with classic statistics that can be net [14,15], but one can also recalculate some ofmanipulated, them. The computer mathematically. used by the author is a personal computer, and therefore certain calculationsObservation sometimes is also take a good an exorbitant complementary tool, preceding a demonstration. This is processing time: the calculation of certain zeroswhat takes we days will of expose calculation by presenting time to obtain. the graphical evolution of the local distribution of con- All the programs were written in Python, withsecutive the graphics zeros and library the evidencematplotlib forand the the presence of only single roots in the CS. mathematical library mpmath to allow for calculations with a number of adjustable significant figures (in the order of 15, 30 or 50, depending2.4. on Data the and case) Computer for floating Science point Tools numbers. Used This study was performed using the set , which represents the first 6 million zeros, 2.5. Mathematical Tools Used as well as a few other sets of higher values. On the interval ≔{0<<10}, we ob- 2.5.1. Stochastic Processes serve, without exception, that the zeros are located on the CL (=½) according to the Yn is a random variable (an RV). The RVs Ytwo-layeredn are pairwise model independent (Lambert RVs. function and stochastic Laplace–Gauss process) described 2 Ten is a Gaussian RV Ten ∼ N µen, eσn . The RVsbelow.Ten areTo obtain pairwise the independent. list of zeros on the CL, one can consult several websites on the Inter- net ([14,15]), but one can also recalculate some of them. The computer used by the author is a personal computer, and therefore certain calculations sometimes take an exorbitant processing time: the calculation of certain zeros takes days of calculation time to obtain. All the programs were written in Python, with the graphics library matplotlib and the mathematical library mpmath to allow for calculations with a number of adjustable significant figures (in the order of 15, 30 or 50, depending on the case) for floating point numbers.

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2.5. Mathematical Tools Used Information 2021, 12, 22 5 of 35 2.5.1. Stochastic Processes

is a random variable (an RV). The RVs are pairwise independent RVs. is a Gaussian RV ~(, ). The RVs are pairwise independent. The two-layerThe two-layer hierarchical hierarchical model modelM is: ℳ is: An anamorphosisAn anamorphosisϕ from C fromto Ce : ℂ to ℂ:

: ⊂ ℂ ⟶ ⊂ℂ ϕ : S ⊂ C → Se ⊂ Ce :M(,: ()x,⟼y) →: (Me, :̃) x=, et (=)ϕ ; (M̃ = ln) ; et =(⁄)t ln;(t/ =e); ⁄t 2= y/2π Information 2021, 12, x FOR PEER REVIEW {Y} {T} n Y L4 of 35 Two stochasticTwo stochastic processes processesand {} andare { adjoined} are adjoined by independent by independent RVs n RVson on associatedℒ with independent Gaussian RVs Ten on Le, n representingℒ the nth zero. associated with independent Gaussian RVs on , representing the nth zero. The pairThe pairYn, Ten , forms forms real RVs real on RVs the on probability the probability space (spaceΩ, A , (PΩ,), , measurable ℙ), measurable functionsfunctions of (Ω ,ofA ()ℬΩ,on=1; ℬ () ,on( ℝ,=−½; >0:ℬ)): ℬ(ℝ): =0 ;ℬ = ;ℬ =− ;ℬ = ;ℬ = R R : ℝ,ℬ(ℝ−) ⟶ ;ℬ ℝ=,ℬ(ℝ;ℬ) =− ;ℬ = ;ℬ =− ;ℬ = ; ϕ : (, ( )) →( , ( )) R R R R ()! ℬ = (−1)|ℬ |; ℬ ~(−1) () ⟼ ~( =, ); 2 = () Yn → Ten ∼ N µen = n, eσn ; Ten = ϕ(Yn) 2.2. Numerical and Graphical Observation as a Tool for Reflection ((y)dy = (t̃)dt̃ == (t̃) lnln tdt = (t̃)(⁄/(22π))lnln((2y⁄)/2π)dy Y Te e e Te e Te e It is fruitful to numerically represent an2d visualize the catalog of formulas, well 1 − 1½( et−µn ) y 1 2 σn known to all, to verify some== exact 3/2formulas⁄e withln infiniteln2π dy sums and to validate approxima- σn((22π) ) 2 tions or inequalities, demonstrated mathematically, to testify to their robustness, speed of t ln (t/e)−n 2 − 1 ( ) convergence and power of (truth.⁄) Graphical2 σvisun alization is a powerful tool for analysis ½ e ln t and reflection, which amplyY( guides2πt) = lnintuition to take paths that are difficult, if not impos- ( ) σ (2π)3/2 2 = ⁄ n sible, to take with the traditional(2) manipula tion of mathematical formulas. However, the We function can consider calculations that thebecome stochastic complicated processes, even are impractical, Markov processes on a computer, (at least when for We7 can consider that the stochastic processes are Markov processes (at least for < n < takesN = 10large): values ( > 10). The window of practicable computer calculations ignores = 10): large numbers. P(Y + = y + |Y ,..., Yn) = P(Y + = y + |Yn); such that y + > yn; P(Y + = y + ≤ yn|Yn) = 0. n 1 n 1(1 =|,…,n 1) =n(1 =|); suchn 1 that >n 1; (n1 = ≤|) =0. 2.3. The Gap between Observation and Demonstration P(Tn+1 = tn+1|T1,..., Tn) = P(Tn+1 = tn+1|Tn); the constraint is tn+1 > tn; P(Tn+1 = tn+1 ≤ tn|Tn) = 0. ( =|,…,Numerical) =( processing=|) ;and the observation constraint is allow for> the; quashing( = of a≤ formula,|) =0 approx-. With this assumption, there is here a dependence between two adjacent RVs. imationWith or inequality, this assumption, with a singlethere is representa here a dependencetion, when between this representation two adjacent does RVs. not conform to the postulate or original intuition. However, the numerical calculation and graph- 2.5.2. The Truncated Gaussian Distribution Density ical2.5.2. visualization The Truncated on the Gaussian first few Distribution million zeros Density do not demonstrate anything. We know theThe fallacy ﬁrst two of the moments hasty general of an RV conclusionsX are theexpectation drawn from and premature the variance, examinations the centered of a phe- The first two moments= [ ] of2 an= RV[ ]= are the2 expectation− ( [ ])2 and the variance, the centered secondnomenon order moment: observedµ overE Xtoo; σsmallV anX interval,E X whileE theX results. The change RV X which or reverse follows over a second order moment: =[]; ∼ N= []2= [ ] − ([]) . The RV which follows a normalwider distribution interval. The is denotedwell-known by: Xexampleµ is, σthe .Skewes number on the difference () − a normal distribution is denoted by: ~(, ). − 2 1 − 1 ( x µ ) (), where the sign swaps for huge values( (a) current= √ upper2 boundσ where this occurs is The Gaussian probability density is: f x ½e ; standard law: 1.39822The× 10Gaussian). For probability the function, density the is: profile () = of theσ 2realπ and imaginary; standard surfaces law: ( has) = 1 − 1 x2 √ ϕ(x) = √ e 2 more2 and½π more tightly packed folds, with increasingly strong amplitudes. The mixing of RVsA√ real and RV theX isrules said for to havethe appearance density if there of zeros exists undergo a positive very and slow integrable evolutions, function so thatf the + A real RV is said to have density if there exists2 a positive and integrableR b function on Robservations, called a density for <10 function, risk such masking that for new all ( landscapesa, b) ∈ R : ofP (thea ≤ X function≤ b) = fora fhigher(t)dt. val- on ℝ , called a density function, such that for all (, ) ∈ℝ : ℙ(≤≤) = ues,For anbeyond RV X ,10 of distribution. A (perhaps function daring)F ,extrapolation of density f , will the conditioning nevertheless athelp the to interval predict the (). [a, b]behaviorgives: beyond =10. TheFor methodology an RV , of distribution here consists function in first presen, of densityting the graphs, the conditioning and statistics at onthe the interval raw [, ] gives: P(a ≤ X ≤ x) F(x) − F(a) Z x x ∈variables.[a, b] : P( XAn≤ analysisx|a ≤ X then≤ b )allows= these variables= to undergo ;geometricF(x) = transformations,f (u)du. P(a ≤ X ≤ bℙ)(F)(b) −(F)()(a) −∞ in order to ∊obtain[, more]: ℙ(≤ abstract|≤≤ calculated) = variables =but with classic;() statistics= ( that) can. be ℙ() ()() manipulated, mathematically. f (x) The associated density is: g(x) = (). The truncated normal distribution X that ObservationThe associated is also density a good is: complementary(F(b))−=F(a) . Thetool, truncated preceding normal a demonstration. distribution This that is we constrainwhat we towill belong expose to by the presenting interval [a ,theb] hasgraphical( the)() density: evolution of the local distribution of con- we constrain to belong to the interval [, ] has the density: secutive zeros and the evidence for the presence of only single roots in the CS. 1 x − µ b − µ a − µ f (x, µ, σ, a, b)=(, ,ϕ , , ) = /φ ; − φ− ; φ distribution distribution function. function. 2.4. Data and Computerσ σ Science Tools Usedσ σ This study was performed using the set , which represents the first 6 million zeros, as well as a few other sets of higher values. On the interval ≔{0<<10}, we observe, without exception, that the zeros are located on the CL (=½) according to the two-layered model (Lambert function and stochastic Laplace–Gauss process) described below. To obtain the list of zeros on the CL, one can consult several websites on the Inter- net ([14,15]), but one can also recalculate some of them. The computer used by the author is a personal computer, and therefore certain calculations sometimes take an exorbitant processing time: the calculation of certain zeros takes days of calculation time to obtain. All the programs were written in Python, with the graphics library matplotlib and the mathematical library mpmath to allow for calculations with a number of adjustable significant figures (in the order of 15, 30 or 50, depending on the case) for floating point numbers.

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2.5.3. The Error Function The error function is used to estimate the percentage of the population which ﬁnds itself inside and outside of a given interval. For the normal distribution, the interval is: µ ± zσ. On Table2, we show, for each interval, the percentage of the concerned population.

Table 2. Conﬁdence interval and % of the population.

Percentage of the Population Inverse of the Probability of Conﬁdence Interval within the an Event Occurring Outside Conﬁdence Interval the Interval µ ± σ 68.3 3.15 =∼ 3 µ ± 2σ 95.45 21.98 =∼ 20 µ ± 3σ 99.73 370.4 =∼ 400 µ ± 4σ 99.993666 15, 787 =∼ 20 × 103 µ ± 5σ 99.99994267 1, 744, 278 =∼ 2 × 106 ± ∼ 1 9 µ 6σ 99.9999998027 506, 797, 346 = 2 × 10 ± ∼ 1 12 µ 7σ 99.999999999744 390, 683, 116, 666 = 2 × 10 µ ± 8σ 99.99999999999988 818, 836, 295, 885, 545 =∼ 1015

√ The proportion of the population in the interval is: P(X ∈ [−z, z]) = er f z/ 2

n 2n+1 3 5 7 9 2 ∞ (−1) z 2 z z z z er f (z) = √ =∼ √ z − + − + + ··· π ∑n=0 n!(2n + 1) π 3 10 42 216

2.5.4. Lambert’s Function Lambert introduced his eponymous function in 1758 [2]. In general, this function is outstanding for compressing the values of mathematical functions that increase even more rapidly than the exponential, and that of functions of related physical phenomena (Figure1).

Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

Lambert’s function enables us to examine the behavior of functions with large numbers, such as π(x) and Li(x). The Lambert function overwrites high values with a more powerful mechanism than that of the logarithm function, by abrogating the exponential component of the function: W(z) = W(wew) = w. In the use we are going to make of it, it will remove the exponential component ew from the function W, applied to w = ln u : W(z) = W ln ueln u = W(u ln u) = ln u. In this article, it allows us to present the distribution of non-trivial zeros on the CL. Lambert’s W function is the reciprocal of the complex variable function deﬁned by f (w) = wew:

l − 1/e ≤ y < 0 : −1 ≤ W(y) < 0 Information 2021, 12, x FOR PEER REVIEW 7 of 35

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≥ > y 0 : W(y) 0 Figure 1. Comparison betweenW the(− Lambert1/e) = − and1; WLogarithm(0) = 0; functions:W(e) = 1;Lambert’s function compresses high values more than the Logarithm function. W0(y) = W(y)/(y(1 + W(y)) 2 2.5.5. Hadamard’s Product Formula L2 L2 For high values of y: L1 = ln y; L2 = ln(ln y); W(y) = L1 − L2 + L + O L . Hadamard’s product formula contains in 1its expression,1 on the one hand, the non- trivial zeros and on the other hand, the functional Riemann equation. It therefore has a 2.5.5. Hadamard’s Product Formula Figure 1. Comparison between the Lambert andremarkable Logarithm strength functions: to Lambert’scharacterize function the zeta compresses function high and values its deep more relationship with neigh- Hadamard’s product formula contains in its expression, on the one hand, the non- than the Logarithm function. borhoods of zeros. We know the formula: trivial zeros and on the other hand, the functional Riemann equation. It therefore has ′() 1 Γ′(1+½) 1 1 a remarkable2.5.5. Hadamard’s strength to Product characterize Formula the= ln2−½−1− zeta function and its deep−½ relationship+ with + neighborhoods of zeros. We know the() formula: −1 Γ(1+½) − Hadamard’s product formula contains in its expression, on the one hand, the non- (½) trivial zeros and on theMoreover, other hand, we know the 0func that:tional ∑ Riemann=1+ equation.⁄ 2−ln2−½ln; It therefore has a = (1+½) 0 + 1 (½) ζ (s) 1 1 1 Γ 1 2 s 1 1 remarkable= strength− toThe −characterize formula− becomes:− the zeta function+ and∞ its deep relationship+ with neigh- ln 2π γ 1 ∑n=1 borhoodsζ(s) of zeros. We2 know thes − 1formula:2 + 1 s s − sn sn (Γ) 1 2 ′ 1 1 ( ) = ln√(−½ ()1+½) − + ′ (1) Γ′ 1+½ −11 1 − = ln2−½−1− −½ +Γ0(1+ 1 s) + InformationMoreover, 2021, 12 we, x know(FOR) PEER that: REVIEW∞ 1 = 1 +−1γ/2 − lnΓ 2(1+½− 1 ln π) ; 2 =−ψ0 1 + 1 s 18 of 35 ∑n=1 sn 2 1 2 Using a CL point of ℒ and the zerosΓ (⊂ℨ1+ 2 s) on the CL ℒ⊂, which assumes the The formula becomes: (½) Moreover, weRH know to be that:true, ∑the infinite=1+ sum⁄ is 2 −ln2−½ln;written: ∑ =−=∑(1+½ ) (½) 0 The formulaζ (s) becomes:We√ alternationfinally 1 obtain, that1 taking generates into1 accoa non-trivialunt∞ the 1pairs zero of in zeros a restricted on the CL interval. (>0 Onand the <0 CL, ):the calcu- = ln π − ψ0 1 + s − + ζ(s) 2 2 s − 1 ∑n=1 s − s ′() lations′ are() extremely simplified:1 1n 1 1 1 = ln −½= ln(1+½√ −½) −(1+½+) + − + √ Using a CL point =½+; of(L)and the( zeros())=Z() ⊂(Ẑ)on= the() CL−1(L)̅ = ⊂ S(, which)̅ (−) ⇒ assumes|()|−=1⇒ the +() = ∞ 1 ∞ 1 1 RH to be true, the infinite sum is written: ∑n=1 = −(i∑) +2n=1 Using a CL point of ℒ and the zeros− ⊂ℨn on the CL− ℒ⊂n , which assumes the Γ(̂⁄)2 Γ(2⁄) − We finally obtain, taking into account the pairs of zeros on the CL (y > 0 and y < 0): RH to be true, the infinite sum is written:() = ∑ ==−∑ ; ()() =1 This formula convergesΓ (slowly2⁄) on theΓ( imaginary2⁄) component, because of the terms in Weζ0( finally) obtain,√ taking into account the pairs of zeros on the CL (>0 and <0): . 1 0 1 1 ∞ 1 1 = ln π − 2 ψ By1 + multiplying2 + − byi ∑ n=(1) if is+ not a zero: () = ()()() =()|()| ζ( ) ′() 1 − n 1 + n 1 (⅝) When is very large,∞ we1 can do the approximation: ≈ ⅝ ≈ . = ln√= f−½( ) + (21+½i ∑ = ) +2 −2 + ( ) n( )1 ℑ() n−̅ − +⁄ |()|The≠0 ⇒ sum can() then= be broken = down ⇒1 into() two= ± sums|()| if ⋘⇒ : =tan = () +2|()| ℜ() 2 This formula converges slowly on the imaginary component, because of the terms 1 − 1 1 −2 = + in n . If is zero, Hospital’s rule gives: () =−()| ( )| This formula converges slowly on the imaginary − component, − because5 of the terms − in 2π(n−2+ 8 ) 2πn When is very large, we can do the approximation: ≈ ≈ . . n ( ) n n−2+ 5 ln n−1 ℜ () We can then choose′ to obtain a good() approximation,8 () ⁄ but the convergence is al- |()| =0 ⇒() =− = ⇒ W = ±e (⅝ ) ⇒− =tan When isways very slow.large, we can do|′ (the )|approximation: ≈ ⅝ ≈ .ℑ () 2 The sum can then be broken down into two sums if ≪ n: If was always very small compared to , which is never the case since ⁄ The sum can then be brokenThe downnon-trivial into two zeros sums are if at⋘ the breaking: point of the two functions + and ∞ sweeps1 the entire⁄ m range1 of ordinates,∞ we coul1d have an estimate of the infinite sum: −= . + ∑n=1 2 2 ∑1 n=1 2 2 1∑n=m+1 2 2 1 − − − n Using = then power1 series1 +expansionn ln of − 1Γ(2⁄) , we serially expand () to find : ⋘ −⇒ ≈− − 4 We can then choose m |to( obtain)| =1⇒ a good() approximation,= =() ; but = the ⁄ 4 convergence−2{̃}; = is −1 al-⁄(24) ; We can then choose to obtain a good1 approximation,1 + 2 + 2 ln but 2 the + convergence() −24ln is always slow. = ways slow. 4 6 If was always very small3i compared9 139i to n, which475 is never the case sincei 1n sweeps73i 131 If was=1− always −very small+ compared+ to , which+⋯1+ is never− the case− since + +⋯ the entire range of ordinates,8 we128 could have3072 an estimate 32768 of the infinite sum:3 18 1620 9720 sweeps the entire range of ordinates, 1 we1 could have an estimate of the infinite sum: ≈ 1 + 2 + 2 ln 2 + ′′(2)−ln 24 i 1 1 1 (6×7×24−5ln − 1 )i 7×24 −5 19 × 31 × 8707i 317 × 531103 ⋘= =1− ⇒ − ≈ − − − − 24 2! (244) 5×3!(24) 5×4!(24) 7×5!(24) 35 × 6! (24) 13 ×1 5243712823i1 + 2 + 231 ln ×2 661 + ×( 28660561) −24ln 27 × 8197117457520809i − = − − 5×7!4 (24) 5×8!6 (24) 25 × 11 × 9! (24) 81 × 5099 × 4517050132181 − +⋯ 1 1 ( ) ∞ 1 11×5×10!1 24 00 ≈ 1≈ + 2(1 ++ 22 lnγ 2+ 2 + ln ′′ 2(π2)+−lnζ (2)) − ln A ∑n=1 242 24 n We can therefore extract the rotation ==−1⁄(24) from this expansion: = + 1() 0 A is the Glaisher–Kinkelin constant: ln A = 12 − ζ (−1) ≈ 0.2487544770337843. = + ()≈=−1⁄(24)

is a three-order polynomial in = −⁄(4! ): 24 17807 29 × 10079 43 × 11009087 13131 × 23027 = 7 − 7 + − + − 5 7×2! 7×5×3! 7×5×4! 7×5! 3458158927391587 7 × 47 × 2095343 × 148000093 + − +⋯ 3×5×7×9×11×6! 3×5×7×9×11×7!

168 168 () = ≈ + (1+)≈() 1 + ≈ ≈() 5 5

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Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

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Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

2.5.5. Hadamard’s Product Formula Hadamard’s product formula contains in its expression, on the one hand, the non- trivial zeros and on the other hand, the functional Riemann equation. It therefore has a Figure 2. Density of non-trivialremarkable zeros onthe strength CL for to the characterize first 6 million the zeros. zeta Thefunction density and of its the deep appearance relationship of zeros, with on neigh- the y axis from 0 to 3 × 106, isborhoods calculated of onzeros. segments We know of length the formula:200. It is verified, on the zoom, that the experimental law remarkably follows the theoretical curve,′( by) observing that the number1 of zerosΓ′(1+½ in a sliding) window, at1 this location1 (1.55 × 106 to 1.6 × 106 ), ranges from 196 to 199.= ln2−½−1− −½ + + () −1 Γ(1+½) − (½) FigureMoreover,3 illustrates we know the that: error ∑ between =1+ the raw⁄ 2 density−ln2−½ln; and the theoretical= density.(1+½) (½) The formula becomes: ′() 1 1 = ln√ −½(1+½) − + () −1 − Using a CL point of ℒ and the zeros ⊂ℨ on the CL ℒ⊂, which assumes the RH to be true, the infinite sum is written: ∑ =−∑ We finally obtain, taking into account the pairs of zeros on the CL (>0 and <0): Figure 3. Error between the theoretical and experimental′() distribution densities. This1 error varies for1 the set D between1 −2 = ln√ −½(1+½) + − + and +2 for segments of length 200. Information( 2021) , 12, x FOR PEER REVIEW ̅ − + 18 of 35 1 = () +2 If the probability density is 1/(2π) ln( /(2π)), the − rank of a particular zero within l m Z ⊂ L ( ) = = R a u u = − = t − the sequenceThis formula convergesis R a slowlyn onalternation0 theln imaginary2π d that2π generates component,2π ln a non-trivial2 πbecausee d of zero thet ln termsine a drestricted in interval. On the CL, the calculations are extremely simplified: . R u u and the number of zeros less than or equal to is the integral N( ) =(⅝ln) 2π d 2π = 0 When is very large, we can do the( approximation:) ( ) ( ) ( ≈) ( )⅝( ≈)() . | ( )| ( ) t 1 arg(ζ(=½+; )) = ̂ = ̅ = ⇒ =1⇒ = t ln e − c; c = −1 + 8 − π ; t = 2π . We highlight here the status of the variable = The( sum) can then be broken= down1 + into two sums if ⋘: t y/ 2π of the ordinate y of 2 iy which comes intoΓ( plaŷ⁄)2 in the zeroingΓ(2⁄) mechanism of the ζ function via the anamorphosed1 variable(1) et== t ln(t/e).= Thus,1 the Lambert; ()(W) =1 = +Γ(2⁄) Γ(2⁄) function appears through the product z ln z present in the integral formula and in many − − − formulas of the ζ function. The Lambert functionBy multiplying thus makes by ( it) possible if is not to exhibita zero: the() = ()()() =()|()| We can then choose to obtain a good approximation, but the convergence is al- anamorphic space Ce by the transformation of the coordinates of C: ways slow. () ℑ() |()| ≠0 ⇒() = = ⇒ () = ±|()|⁄ ⇒ =tan If nwas always very small compared to | (, which)| is never1+W (thet/e) ocase since x → x; y → et = ϕ(y) = y/(2π) ln(y/(2πe)) ⇔ y = 2πe e . ℜ() 2 sweeps the entire range of ordinates, we could have an estimate of the infinite sum: If is zero, Hospital’s rule gives: () =−()| ()| The passage into a transformed 1 1 space vialn this − function 1 makes it possible to simplify the study⋘ of the ζfunction.⇒ After≈ the anamorphosis of the y axis, in Figure4, we determine 4 ′() ℜ () 5|()| =0 ⇒() =− = ⇒ () = ±()⁄ ⇒− =tan the constant d = −2 + , by fitting the line et = φ(n). () 8 1 1 + 2 + 2 ln 2|′ +( )| −24ln ℑ () 2 = 4 6 The non-trivial zeros are at the breaking5 point of the two functions +⁄ and et = ϕ(y) = y/(2π) ln(y/(2πe)) = t ln t/e = φ(n) = n + d = n − 2 + . 1 1 −⁄ . 8 ≈ 1 + 2 + 2 ln 2 + ′′(2)−ln Using the power series expansion of Γ(2⁄), we serially expand () to find : 24 |()| =1⇒() = =() ; = ⁄ 4 −2{̃}; = −1⁄(24) ;

3i 9 139i 475 i 1 73i 131 =1− − + + +⋯1+ − − + +⋯ 8 128 3072 32768 3 18 1620 9720

We can therefore extract the rotation ==−1⁄(24) from this expansion: = + ()

= + ()≈=−1⁄(24)

168 168 () = ≈ + (1+)≈() 1 + ≈ ≈() 5 5

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Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

1 1 Figure 5. (a) Histogram of et with 500 intervals: frequency minimum is at 1/8, maximum is at 5/8. We recognize ≈ 1 + 2 + 2 ln 2 + ′′(2)−ln a histogram made up of several random variables folded on themselves and with24 an average of 5/8; (b) histogram ε = R a − n.

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Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

1 1 ≈ 1 + 2 + 2 ln 2 + ′′(2)−ln 24

6 Figure 6. Error ε = R a − n = ±1 versus n; n ≤ 6 × 10 .

In Table3, we present the list of the 50 first zeros which are classified with an error of −1 and an error of +1, i.e., the list of the first zeros which are, behind and ahead, respectively.

Table 3. The 50 ﬁrst early and late zero numbers.

Early Zero Number (ε−1) Late Zero Number (ε+1) 136 705 1096 1331 1671 127 596 820 1179 1488 213 779 1113 1346 1686 196 619 849 1210 1496 256 795 1137 1404 1704 233 627 858 1251 1499 379 809 1158 1445 1746 289 655 888 1277 1514 399 871 1167 1487 1762 368 669 965 1291 1546 509 994 1209 1519 1789 380 693 996 1308 1614 531 1018 1233 1534 1798 401 716 1029 1332 1621 580 1048 1265 1545 1806 462 729 1035 1343 1647 639 1073 1296 1602 1821 519 767 1044 1431 1657 696 1088 1321 1645 1858 568 796 1114 1457 1673

When we know the ordinates of the zeros, we can therefore rectify the incorrectly estimated zeros in the formula and restore the residual which is greater than 1 in absolute value. Once this correction Ten → Uen = Ten − (±ε) has been carried out, one can calculate the new histogram which is plotted on Figure7. One deduces the second layer of the theoretical model from it. A zero is the realization of a Gaussian random variable Uen. We can −(± )− deﬁne the standard Gaussian variable V ∼ N (0, 1) such that: ∀n : V = Ten εn µn and e en σn estimate the probability density function (PDF) which turns out to be almost a perfect Gauss Law. Information 2021, 12, x FOR PEER REVIEW 7 of 35 Information 2021, 12, 22 12 of 35

Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

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Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

This formula converges slowly on the imaginary∗ component, because ofΓ (thê⁄)2 terms5 inΓ (2⁄) ∀n : n = 2πt5,n/W(t5,n/e);(t5,)n== n− 2 + .= ; ()() =1 . Γ(2⁄)8 Γ(2⁄) (⅝) ≈ ≈ When is very large, we can do the∗ approximation:10 ⅝ . The error is: n − n < . See FigureBy9 multiplying. by () if is not a zero: () = ()()() =()|()| ln n The sum can then be broken down into two sums if ⋘: () ℑ() 1 1|()| ≠0 ⇒ () =1 = ⇒ () = ±|()|⁄ ⇒ =tan |()| ( ) 2 = + ℜ − − − ( ) ( )| ( )| We can then choose to obtain a good approximation,If but the is zero, convergence Hospital’s is rule al- gives: =− ways slow. ′( ) ℜ() |()| =0 ⇒() =− = ⇒ () = ±()⁄ ⇒− =tan If was always very small compared to , which is never the case since |′()| ℑ () 2 sweeps the entire range of ordinates, we could have an estimate of the infinite sum: The non-trivial zeros are at the breaking point of the two functions +⁄ and 1 1 ln − 1 ⋘ ⇒ ≈ −⁄ . 4 ∗ Γ(2⁄) () () Using the power series expansion of , we serially expand to find : Figure 9. Scatter plot n − n 1ln n1versus + 2 +. 2 Errors ln 2 become + smaller−24ln and smaller as a function of . = () 4 6 |()| =1⇒() = = ; = ⁄ 4 −2{̃}; = −1⁄(24) ; Figure 10 shows a sequence of zeros, on the transformed y axis, with the regularly 1spaced1 . 3i 9 139i 475 i 1 73i 131 w5 ( ) ≈ 1 + 2 + 2 ln 2 + ′′=1−2 −ln− + + +⋯1+ − − + +⋯ 24 8 128 3072 32768 3 18 1620 9720

i 1 (6×7×24−5)i 7×24 −5 19 × 31 × 8707i 317 × 531103 = =1− − − − − − 24 2! (24) 5×3!(24) 5×4!(24) 7×5!(24) 35 × 6! (24) 13 × 5243712823i 31 × 661 × 28660561 27 × 8197117457520809i − − − 5×7!(24) 5×8!(24) 25 × 11 × 9! (24) 81 × 5099 × 4517050132181 − +⋯ 11×5×10!(24) We can therefore extract the rotation ==−1⁄(24) from this expansion: Figure 10. The 2-layer model: in the middle of the ﬁgure, two zeros (n = 4, 088, 664, 936, 218) and = + () n + 1 are very close (∆y = 1.71 × 10−5 ), one late, the other early, respectively. The random realization of these two zeros almost falls on t = 1 , which may seem a paradox, since it is for this value 1 , e 8 = + ()8≈=−1⁄(24) on this anamorphosed axis, that the probability of occurrence is the lowest to get a zero. is a three-order polynomial in = −⁄(4! ): When n is large, the formula becomes: 24 17807 29 × 10079 43 × 11009087 13131 × 23027 = 7 − 7 + − + − 55 7×2! 7×5×3! 7×5×4! 7×5! 2π n − 2 + 8 2πn 2πn 2πn n n → ∞ ⇒ n ≈ ≈ n 3458158927391587≈ n n ≈ 7 ×≈ 472 ×π 2095343 × 148000093 n−2+ 5 − lnn − 1 ln n 8 W+e ln e ln ln e − +⋯ W e 3×5×7×9×11×6! 3×5×7×9×11×7!

We undoubtedly ﬁnd the similarity 168 with the formula of the prime number theorem of = 1 + n Hadamard and La Vallée Poussin (1896):5 π(n) ∼ ln n . 4 36 + 31 × 12 + (34 × 239 + 2 × 1643717 + 53 × 66797) 3.3. The Heteroscedasticity of Gaussian RVs 35 5 1 Figure 11 shows the adjustment of the variation of the standard deviation, calcu- + (68 × 61 × 107923 × 965611 + 23 × 4881469 × 16227313 ) +⋯ lated by an estimate on a sliding neighborhood45 11 of 1000 zeros. After a rapid increase in the ﬁrst values, there is a slow increaseWe in thecan variance.deduce an The approximate experimental value law of is:() on the CL: eσn = 0.12555 W(ln n) + 0.08856. 168 ≈ (1+); = −1⁄(24) 5

168 168 () = ≈ + (1+)≈() 1 + ≈ ≈() 5 5

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Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

2.5.5. Hadamard’s Product Formula Hadamard’s product formula contains in its expression, on the one hand, the non- trivial zeros and on the other hand, the functional Riemann equation. It therefore has a remarkable strength to characterize the zeta function and its deep relationship with neighborhoods of zeros. We know the formula: Information 2021, 12, 22 14 of 35 ′() 1 Γ′(1+½) 1 1 = ln2−½−1− −½ + + () −1 Γ(1+½) − (½) Moreover, we know that: ∑ =1+ ⁄ 2−ln2−½ln; = (1+½) (½) The formula becomes: ′() 1 1 = ln√ −½(1+½) − + () −1 − Using a CL point of ℒ and the zeros ⊂ℨ on the CL ℒ⊂, which assumes the Information 2021, 12, x FOR PEER REVIEW 18 of 35 RH to be true, the infinite sum is written: ∑ =−∑ We finally obtain, taking into account the pairs of zeros on the CL (>0 and <0): Figure 11. Adjustment of the variation of the standard deviation according to n. ′() alternation that generates1 a non-trivial zero1 in a restricted1 interval. On the CL, the calcu- = ln√ −½(1+½) + − + () ̅ − + 3.4. The Equivalence Relation on the CL lations are extremely simplified: 1 = () +2 The zeros of the real and imaginary parts of ζ(=½+; ) are close to() the= pivots()(̂) =w5,n()()̅ =( )() ⇒ |()| =1⇒() = − and w1,n: This formula converges slowly on theΓ (imaginarŷ⁄)2 component,Γ(2⁄) because of the terms in () = = ; ()() =1 1 1. Γ(2⁄) Γ(2⁄) ∀n : ζ + i w5,n ≈ 0 + ibn; ζ + i w1,n ≈ an + i × 0; an, bn ∈ R (⅝) 2 2 When is very large, we can do the approximation: ≈ ⅝ ≈ . By multiplying by () if is not a zero: () = ()()() =()|()| 5 The sum can 1then be broken down into two sums if ⋘: t5,n = n − 2 + ; w5,n = 2πt5,n/W(t5,n/e); t1,n = n − 2 + ; w1,n = 2πt1,n/W((t)1,n/e). ℑ() 8 |(8)| ≠0 ⇒ (1) = =1 ⇒ ()= ±|()|1⁄ ⇒ =tan |()| ℜ() 2 The alternating and regular setting to zero of the real and imaginary parts= of theζ + − − − function on the CL in the neighborhoods of the pivot values w5,n for the real part and ( ) ( )| ( )| We can then choose If to obtain is zero, a Hospital’sgood approximation, rule gives: but the convergence=− is al- w1,n for the imaginary part, are at the origin of the production of the nth zero of the ζ ways slow. Information 2021, 12, x FOR PEER REVIEW function, a zero which this time is common to both real and3 of imaginary 35 parts.′ These() pivot ℜ () |()| =0 ⇒() =− = ⇒ () = ±()⁄ ⇒− =tan If was always very small compared to , which is never the case since values are not exactly the points of nullity for which a precise expression is later|′( calculated.)| ℑ () 2 However, at pivotal values, there is the relation:sweeps the entire range of ordinates, we could have an estimate of the infinite sum: +⁄ The non-trivial zeros are at the breaking point of the two functions and Table 1. Notations. 1⁄ 1 ln − 1 1 ⋘ ⇒− . ≈ ∀n : ζ + i w5,n ≈ an(1 + i48yw5,n) 4 Definition Notation 2 Using the power series expansion of Γ(2⁄), we serially expand () to find : 1 1 + 2 + 2 ln 2 + () −24ln ∀ ∊ ℝ: || = > 0; − < 0 = () Absolute value |()| =1⇒ () = = ; = ⁄ 4 −2 {̃}; = −1⁄(24) ; 1 () 4 6 ∀ ∈ ℂ: = + =∀ n : ζ = |+|i w1,n ; ≈ bn(−48yw1,n + i) 2 || = |+| = + ; , , , ∈ ℝ 1 3i 1 9 139i 475 i 1 73i 131 =1− ≈ − 1 + 2+ + 2 ln 2+ ′′(2)−ln+⋯1+ − − + +⋯ Complex number 7 = (⁄) 2 |an|, |bn| < ε, >ε 0;< 1/n; pour n <810 24. 128 3072 32768 3 18 1620 9720 = (⁄) + < 0; =0: =On the CL,⁄ 2 there > is 0; a relationship = −⁄ 2 between < 0 the real andi imaginary1 parts(6×7×24−5 of the )i 7×24 −5 19 × 31 × 8707i 317 × 531103 = =1− − − − − − In ℝ, theζ function:integer part and the fractional part of a real number : Integer and fractional parts 24 2! (24) 5×3!(24) 5×4!(24) 7×5!(24) 35 × 6! (24) ∀ ∊ ℝ: =[] + {}; [] ∊ℕ,0≤{} <1 13 × 5243712823i 31 × 661 × 28660561 27 × 8197117457520809i 1 1 − − − Complex number in the CS In ℂ, =+ =ℜ∀ = (+)i + ℑ: (ζ)(;∈ℂ;,∊))/R(ζ( ))ℝ=; tanits conjugateω; α = π /4̅ =− − 2π et ; θ = 1/((24 ); ω ≈ α + θ ( ) ( ) 2 2 5×7! 24 5×8! 24 25 × 11 × 9! 24 In ℂ, ̂ =1−=1−− ;,∊ℝ; 81 × 5099 × 4517050132181 Complex number 1− associated with − +⋯ itsThis conjugate relation (demonstrateď =1− =̂̅ =1−+ later in Section. 3.6.2) is only valid11×5×10! on the CL,( and24 takes) into account an angle related to the fractional part of the ordinate et, which allows ̂ symmetric of with respect to (½, 0). ==−1⁄(24) us to anchor the common zeros of these two components in an equivalenceWe can therefore relation, extract the rotation from this expansion: =ℜ() + ℑ() = + ; 0≤≤1,>0; ≠1 Critical strip (CS) In the CS : . = + () 1 ≡ 2 ⇔ et1 ≡ et2 ⇔ et1 = et2 : Critical line (CL) On the CL ℒ∶==½: =½+; >0. = + ()≈=−1⁄(24) \ ℒ∶=0≤≤1;≠½; ≠1∼ ∼ CS, except the CL 1 ≡ 2 ⇔ α1 = α2 ⇔ ω1 = ω2 ⇔ (ζ( 1))/R(ζ( 1)) = (ζ( 2))/R(ζ( 2)). () = ∑ . In the CS, the function is divergent. ⁄( ) Zeta function: is a three-order polynomial in = − 4! : The two alternately zero ζ surfaces of period 1( on) the anamorphic t axis of the ordinate One considers in this CS, the analytical continuation . e of the complex plane locally generate an additional common24 zero which17807 is the nth29 non- × 10079 43 × 11009087 13 131 × 23027 Êta Dirichlet function: () =()(1−2 ). = 7 − 7 + − + − trivial zero. For a non-trivial zero, the ratio becomes50/0, and Hospital’s7×2! rule applies7×5×3! 7×5×4! 7×5! Derivative of the function () =−∑ log()⁄ in : 3458158927391587 7 × 47 × 2095343 × 148000093 C () + − +⋯ () = ∑ −log() 0 0 k-order derivatives of the function (ζ( 1))/R(ζ( 1)) = −R ζ ( 1) / ζ ( 1) = tan3×5×7×9×11×6!(ω/2) 3×5×7×9×11×7! In the CS, one uses the analytical continuation of these derivatives This equivalence() =() relation(̂) =( in)Ce(,1− where) the n classes are intervals168eIn of length 1 (Figure 12), h i = 1 + Functional equation of the function for the segments ∀n : I = n − 1 , n + 1 Γ(characterizeŝ⁄)2 the silhouettes of the ζ function () =2 sin(⁄) 2en Γ(1−)2=½2 5 Γ(2⁄) 4 36 on the CL, proﬁles which extend into the CS, and which allow+ us to31 estimate × 12 + the max-(34 × 239 + 2 × 1643717 + 53 × 66797) () =()(̂) =()()̅ =()(̅ ); 35 5 Functional equation on the CL Γ(2⁄) 1 () = + (68 × 61 × 107923 × 965611 + 23 × 4881469 × 16227313) +⋯ Γ(2⁄) 4511 t=⁄(2) = ⁄ 2 We can deduce an approximate value of () on the CL: ̃ =ln(⁄)=⁄(2) ln(⁄)(2) . The anamorphosis produces a stronger Anamorphosis: ̃ 168 elongation of the axis as increases. ≈ (1+); = −1⁄(24) 5 We define by the adjective “digressive” a series that exhibits a beginning similar to a known series, but which varies noticeably with the subsequent terms. 168 () 168 () ( ) = ≈ + (1+)≈ 1 + ≈ ≈ For example, the cosine series is written: cos = 1 − + − +⋯+ 5 5 ! ! ! Digressive series (−1) +ℴ(). We will then note, in order to simplify the writing, a ()! digressive cosine: () =1− + − +⋯+(−1) + ! ! ! ()! ℴ().

Complex zeros ℨ in the CS = + ; () =0 =½+ ; () =0 Zeros ⊂ℨ on the CL ℒ⊂ The zeros are ordered in pairs, and 1−. Calculations are carried out with >0 Lambert function The main branch is defined by: () = ⇔=;≥−1⁄ . ( ) Gamma function Γ = Euler’s constant Euler–Mascheroni constant: = lim− ln + ∑ ≈ 0.5772156649 → ()() Taylor series () = ∑ (−) ! Bernoulli numbers Coefficients of the power series of = ∑ ℬ !

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ima of |ζ(s)| in this CS. When one takes into account the raw values of the y axis in C, the equivalence relation is more difﬁcult to manipulate and the underlying RVs are more convoluted. The Ce-space distribution model removes the statistical bias caused by the strict y axis support that hides the true mechanisms of the ζ function.

Figure 12. The equivalence relation: a few segments of the ζ curve (real and imaginary parts) along a few intervals of the 5 1 anamorphic et CL. The real (resp. imaginary) parts almost cancel each other out for the values et ≡ 8 , (resp. 8 ). We almost 3 7 also veriﬁed R(ζ(s)) = −I(ζ(s)), (resp. +I(ζ(s))) for the values et ≡ 8 , resp. 8 .

3.5. The Continuous Catalog of Fold Appearances of the ζ Function in the CS The stochastic process contributes to characterizing the local ζ function landscapes. The classification of the ζ function conformations simply takes its source in the effective “random” distribution of the zeros. The key to determining taxonomy is the value of the probability density of the nearby zero (Figure 13). We can thus extract some canonical patterns, at the origin of the production of a non-trivial zero, although the catalog of these patterns constitutes a continuous spectrum. These different families of patterns show more or less regular and more or less high wave configurations, according to the standard pattern: the higher the value of the probability density of a zero is, the more regular the neighboring undulations will be; the lower the value of the probability density of a zero is, the more irregular the nearby waves will be, in particular with weak waves and a strong wave which will appear in the vicinity (Figure 14). When the probability is very low, one can identify a particular archetype consisting of a rogue wave surrounded by dead calm. More frequently, the occurrence of small distances between two consecutive zeros arises with a similar configuration of the ζ function: it is a high wave of the real part, located upstream or downstream, accompanied by wavelets of the real and the imaginary parts which graze the zero level of the plane z = 0: the corrugated sheet of both ζ surfaces in the CS also gives rise to a kind of dead calm, any homothety of the Riemann functional equation apart.

Figure 13. Classiﬁcation of the local behavior styles of the zeta function according to the value of the pdf of the local zero realization. Local styles of the zeta function (real, imaginary and module) are shown: rogue wave style on the left; stationary style on the right. Information 2021, 12, 22 16 of 35

Figure 14. Six different patterns depending on the pdf of the zero linked to the segment In.

3.6. The Structural Characters of ζ and ξ from the Functional Equation In this paragraph, we develop the ξ function in the power series, by taking advantage of the power series expansion of the Gamma function. This operation uses the fact that x is very small compared to y. The calculations are geometrically interpreted in order to identify the structural features of the ζ and ξ functions which are expressed by this functional equation. In the CS, the ξ function is a composition of elementary geometric functions: a homo- 2 1 thety and four rotations θ1 + θ2 + θ3 + θ4, and a series remainder function in x − 2 /y:

∀s ∈ S : ζ(s) ≡ ζ(sˆ) modulo (H ◦ R1 ◦ R2 ◦ R3 ◦ R4 ◦ E)

2 1 −x iω 1 −x i(θ +θ +θ +θ ) 1 ζ(s) = ξ(s)ζ(sˆ) = t 2 e ζ(sˆ) = t 2 e 1 2 3 4 (1 + ε(x − ) /y) × ζ(sˆ) ; ε ∈ C 2 ξ(s) × ξ(sˆ) = 1 ⇒ (1 + ε)(1 + εˆ) = 1 The homothety is a function of x and the four rotations are functions of y. The remainder (1 + ε) is the only factor that mixes the two coordinates x and y. The homothety plays 1 1 its role of ζ ampliﬁcation in the 0 < x < 2 region and contraction in the 2 < x < 1 region. InformationThe four rotations,2021, 12, x FOR two PEER of which REVIEW are pure and the other two are digressive, that is to say a 18 of 35

little falsiﬁed, ensure the functional equation between ζ(s) and ζ(1 − s). On the CL, the homothety is neutral and only the ﬁrst three rotations are effective: the ξ function ensures the relationalternation between thatζ andgenerates its conjugate, a non-trivial via a rotationzero in a of restricted the angle interval. On the CL, the calcu- 3 θ1 + θ2 + θ3, within an order oflations accuracy are extremely of (1/y) : simplified: ∀=½+; ∈ L : ζ( ) ≡(ζ)(=) modulo()(̂)(=R1(◦ R)2(◦)̅ R=3 ◦( E)() ⇒ |()| =1⇒() =

( + + ) 3 ζ( ) = ξ( )ζ( ) = eiωζ( ) = ei θ1 θ2 Γθ3((̂⁄)12+ ε(1/y)Γ()2×⁄)ζ( ); ε ∈ C () = = ; ()() =1 Γ(2⁄) Γ(2⁄) ξ( ) × ξ( ) = 1 ⇒ (1 + ε)(1 + ε) = 1 By multiplying by () if is not a zero: () = ()()() =()|()| 3.6.1. The Functional Equation in the CS The functional relationship is written: () ℑ() |()| ≠0 ⇒() = = ⇒ () = ±|()|⁄ ⇒ =tan |()| ℜ() 2 s− 1 Γ(sˆ/2) s = x + iy; sˆ = 1 − s; ζ(s) = ξ(s)ζ(sˆ) ; ξ(s) = π 2 ; ξ(s)ξ(sˆ) = 1 If is zero, Hospital’s ruleΓ(s /2gives:) () =−()| ()| ′( ) ℜ() |()| =0 ⇒() =− = ⇒ () = ±()⁄ ⇒− =tan |′()| ℑ () 2 The non-trivial zeros are at the breaking point of the two functions +⁄ and −⁄ . Using the power series expansion of Γ(2⁄), we serially expand () to find : |()| =1⇒() = =() ; = ⁄ 4 −2{̃}; = −1⁄(24) ;

3i 9 139i 475 i 1 73i 131 =1− − + + +⋯1+ − − + +⋯ 8 128 3072 32768 3 18 1620 9720

We can therefore extract the rotation ==−1⁄(24) from this expansion: = + ()

= + ()≈=−1⁄(24)

168 168 () = ≈ + (1+)≈() 1 + ≈ ≈() 5 5

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Information 2021, 12, 22 17 of 35

FigureInformation 1. Comparison 2021, 12, x betweenFOR PEER the REVIEW Lambert and LogarithmInformation functions: 2021 Lambert’s, 12, x FOR function PEER REVIEW compresses high values more 17 of 35 4 of 35 than the Logarithm function. The power series of the Gamma function is: 2.5.5. Hadamard’s Product Formula The power series of the Gamma function is: √ 1 ℬ =1; ℬ =−½; >0:ℬ =0 ;ℬ = ;ℬ =− ;ℬ = ;ℬ = Hadamard’s product formulaz− contains−z+µ(z) in its expression, on the one∞ hand, the2k non- Γ(z) = 2πz 2 e pour z ∈ C\Z−; µ(z) = ∑k=1 ℬ 2k−1 trivial zeros and on the other hand,½ the (func) tional Riemann equation. It therefore2k(−2k −;ℬ 1has)z = a ;ℬ =− ;ℬ = ;ℬ =− ;ℬ = ; Γ() = √2 ∈ ℂ\ℤ;() = remarkable strength to characterize the zeta function and its deep relationship 2(2 with − 1 )neigh- ()! ℬ = (−1)|ℬ |; ℬ ~(−1) i are the Bernoulli numbers. () borhoodsℬ of are zeros. the Bernoulli We know numbers. the formula: In the CS, we can then develop the ξ function in series, assuming 0 < x < 1 ≪ y: ′In( the) CS, we can then develop1 the Γ′( 1+½ function) in series, assuming1 1 20<<1⋘: 2.2. Numerical and 1Graphical Observation as a Tool for Reflection t = y/(2π); t = t ln(t/= ln2−½−1−e); θ = π/4; θ = −2π−½t ; θ = −1/(24+y); θ = − x −+ / (2y); ẽ () 1 2 −1̃e 3Γ(1+½) 4 − = ⁄(2) ; =ln(⁄); =⁄ 4; =−2{}; =−1⁄(24) ; It =−(−½)is fruitful 2to⁄ numerically(2) ; represent and visualize the catalog of formulas, well known to(½ all, to) verify some exact formulas with infinite sums and to validate approxima- Moreover, we know that: ∑1 =1+ ⁄ 2−ln2−½ln; = (1+½) ̃ (½) =α = θ1 ++ θ2 ==22π(⅛−−{})et ; ≈; ω ≈ θ+1 + θ+2 +tionsθ+3 + θor4 inequalities, demonstrated mathematically, to testify to their robustness, speed of The formula becomes: 8 convergence and power of truth. Graphical visualization is a powerful tool for analysis ½1 − ½1 − ½1 − ((+) ) ((+ )) ½1 − ((+ )) () () = 2 x iω= 2 x ′i α(()) = 2 x i θ1 θ2 i θ3 θ4 + 1()=and reflection,2 x 1i θ1 θwhich2 i amply guides intuition to take paths that are difficult, if not impos- ξ(s) = t e = t e A(s= ln) = t√ −½e (1+½e ) −+ K(s+) = t e e () −1 sible, to −take with the traditional manipulation of mathematical formulas. However, the θ 3 1 2 function calculations become complicated, even impractical, on a computer, when K(s) = −2iθ4 + 2y x − 2 1 − 10ℒx+ 8x ⊂ℨ ℒ⊂ () =−2 +Using(−½ a CL)( point1 − 10 of + 8 and) the zeros on the CL , which assumes the 2Information 2021, 12, x FOR PEER REVIEW 2 takes large values ( > 10 ). The window of practicable computer18 of 35 calculations ignores RH to be− itrue,θ3 − the7 infinite+ x − sum1 −is 5written:− 1 x + ∑23x − 1 x=−3 + 1∑x4 − x5 y2 120 2 24 2 12 2 2 7 5 23 large numbers. − θ2− + (−½)− −½+ 2−½ 3+½4− We+ finally 3 − obtain,7 + taking− 1 into133 acco+ 2753untx +the707 pairsx − of181 zerosx − on87x the− CL5 (+>06 and+ <0··· ): y2 120120 x 2 72024 360 1260 10 10 5x 12x ′() alternation1 that generates 2.3. aThe1 non-trivial Gap between1 zero Observation in a restricted and Demonstrationinterval. On the CL, the calcu- 7When the point133s is( on2753 the CL,) 707 the expression181 is87 simplified: + − += ln(−½√)−½ +1+½++ −− − +−5 +12 +⋯ 120() 720 lations360 are extremely60̅ 10 simplified:−Numerical10 + processing and observation allow for the quashing of a formula, approx- 1 imation or inequality, with a single representation, when this representation does not con- = ()1+2 168 3 K(=½+; ) = K + i() == i()θ((̂)1=+ θ(3+)···()̅ );=θ3(=)−(1/)(⇒24|y)()| =1⇒() = When the point is on the CL,2 the expression 5 − 3is simplified:form to the postulate or original intuition. However, the numerical calculation and graphical visualization on the first few million zeros do not demonstrate anything. We know This formula converges slowly on the imaginary168 component,Γ(̂⁄)2 becauseΓ(2⁄) of the terms in 1 −x i( + ) i( + ) The serial expansion() = (½+ of ξ(s))==(t2) =eθ1(1+θ2 thee +⋯ θfallacy3=θ4); + ofK =−1the(s) hasty; ⁄is((24 interpreted )general)( ) =1 conclusions as a func- drawn from premature examinations of a phe- . 5 Γ(2⁄) Γ(2⁄) 1 −x nomenon( ⅝observed) over too small an interval, while the results change or reverse over a Whention composed is very large, of awe homothety can do thet 2 approximation:½and four( rotations,) (≈ the) ⅝ first two≈ being. pure rotations. The serial expansion of () = wider() interval.+( The) is well-known interpreted() example = as a( ) (is )the( Skewes) =( )|number()| on the difference () − By multiplying−x by if is not− sa zero: • The homothety supports the n½effect of the() power, where of the then signterms swaps of the forζ hugefunction, values (a current upper bound where this occurs is Thefunction sum can composed then be broken of a homothety down into two sums and four if ⋘ rotations,: the first two being pure ro- with the symmetry at x = 1 . The CL( is) a neutral1.39822 element× 10 of). the For homothety. the function,ℑ( the) profile of the real and imaginary surfaces has tations. |(1)| ≠0 ⇒2() =1 = ⇒ 1() = ±|()|⁄ ⇒ =tan • • The rotation R(θ1 ==π/4) comes from|(+)| the offsetmore ofand the more real tightly surface packed which is,folds,ℜ in prob-( with) increasingly2 strong amplitudes. The mixing of The homothety supports− the effect− of the power of −the terms of the func- ability, above the imaginary surface, due toRVs the and imbalance the rules offor the the first appearance term of theof zeros undergo very slow evolutions, so that the tion, with the symmetry at =½. The CL is a neutral element of the homothety. If is zero, Hospital’s rule gives: ( ) =−( )| ( )| We• can thenseries choose of ( cos (toy ln obtain 1) = 1;a singood(y lnapproximation, 1) = 0). Thisobservations rotation but the restores,convergence for <10 on average, is risk al- masking the two new landscapes of the function for higher val- The rotation ℛ( =⁄) 4 comes from the offset of the real surface which is, in prob- surfaces to the same level. ways slow. ′( ) ues, beyond 10 . A (perhaps daring)ℜ extrapolation() will nevertheless help to predict the ability, above the imaginary surface, due to the imbalance ( )of the first( )term⁄ of the se- If was• Thealways major very|( rotation )|small=0 ⇒R comparedθ2(=) −=−2 πto et=, −which2πbehavior{=t lnis (nevert ⇒/e) beyond } theerases= ± case =10 the since sinuousness . ⇒− of the =tan ries of (cos(ln1) =1;sin(ln1) =0)|′(. This)| rotation restores, on average, the two ( ) 2 ξ function. This rotation, dependent on y, shears by continuous torsions, in orderℑ to sweeps the entiresurfaces range to theof ordinates, same level. we could have an estimateThe of the methodology infinite sum: here consists in first presenting the graphs and statistics on the raw make the undulations disappear. ⁄ • The non-trivial zerosvariables. are at An the analysis breaking then point allows of thesethe two variables functions to undergo + geometricand transformations, The major rotation1 ℛ1( =−2 ln{ ̃} −=−2 1 {ln(⁄)}) erases the sinuousness of the ⋘• The⇒ rotation R≈(θ3 =−−1/⁄( 24. y)) is numerically in order and to graphically obtain more barely abstract perceptible. calculated variables but with classic statistics that can be function. This rotation,4 dependent on , shears by continuous torsions, in order to It is impure because the seriesUsing contains the power higher seriesmanipulated, order expansion terms mathematically. of which Γ(2⁄) alter, we the serially rotation. expand () to find : make the undulations disappear. () This rotation1 has a1 decisive + 2 + 2influence ln 2 + in the−24ln genesis of the zeros of the two surfaces, • = ()Observation is also a good complementary tool, preceding a demonstration. This is The rotation ℛ( =−1|()|⁄)(=1⇒24) is( numerically) = = and graphically ; = ⁄ 4 −2 barely{̃} ; perceptible. = −1⁄(24 ) ; and consequently4 in the materialization6 of non-trivialwhat we will zeros. expose by presenting the graphical evolution of the local distribution of con- It is impure because the series contains higher2 order terms which alter the rotation. 1 •This1The rotation impure1 has rotation a 3idecisiveR 9 θinfluence4 = −139ix −in the /genesis(secutive4752y) isof notzerosthe perceptiblezeros and ofthei the evidence on two1 the surfaces, CL. for73i Itthe has presence131 of only single roots in the CS. ( ) 2 ≈ 1=1− + 2 + 2− ln 2 + ′′+ 2 −ln + +⋯1+ − − + +⋯ andan consequently intrinsic24 role 8in in the the128 materialization CS, because3072 it isof the non-trivial32768 impurity zeros. of this digressive3 18 rotation1620 which 9720 • 2.4. Data and Computer Science Tools Used Thebreaks impure the rotation symmetry ℛ( at =− the level(−½ of) each⁄)(2 term) is of not the perceptible series in 1/ ony and the CL. which It has does an not intrinsicmake role it possible in the toCS,i manufacture because1 it is the the(6×7×24−5 two impurity pairsof Thisof zeros) thisi study digressive7×24 (s, sˆwas and −5performeds, rotationsˆ) outside19 × usingwhich 31 the × 8707ithe CL set 317, which × 531103 represents the first 6 million zeros, = =1− − − − − − breaks(for ythe> symmetryand y <240). at the2! (level24 )of each5×3! term(as 24of well the) asseries a5×4! few in other1(24⁄ and )sets which of7×5! higher does(24 values. ) On35 ×the 6! interval(24) ≔{0<<10 }, we ob- not make it possible to13 manufacture × 5243712823i the two31 pairs ×serve, 661 of ×without zeros 28660561 ( ,exception, ̂ 27̅, ×̅ ) 8197117457520809ithat outside the zerosthe are located on the CL (=½) according to the The third and fourth− rotations are impure− (or digressive) rotations:− the symmetry in x two-layered model (Lambert function and stochastic Laplace–Gauss process) described andCL1 (for− x is> thus and not <0 respected).5×7! by(24 the series) K(s) from5×8! the(24 term) in 1/y2 in25 the × CS, 11 except × 9! (24 on ) 81 × 5099 × 4517050132181below. To obtain the list of zeros on the CL, one can consult several websites on the Inter- theThe CL, third which and geometrically fourth− rotations explains are impure the RH. (or digressive)+⋯ rotations: the symmetry in net ([14,15]), but one can also recalculate some of them. The computer used by the author and 1− is thus not respected11×5×10! by the series(24 )() from the term in 1⁄ in the CS, is a personal computer, and therefore certain calculations sometimes take an exorbitant except3.6.2. on The the Functional CL, which EquationgeometricallyWe on thecan explains CL therefore the RH.extract the rotation ==−1⁄(24) from this expansion: processing time: the calculation of certain zeros takes days of calculation time to obtain. In this paragraph, we show,= using+ the() properties of the functional equation, that the All the programs were written in Python, with the graphics library matplotlib and the 3.6.2.real The and Functional imaginary Equation curves of on the theζ function CL regularly and alternately cancel each other out, mathematical= library + mpmath()≈=−1 to allow⁄ (for24 calculations) with a number of adjustable signifi- thanksIn this to paragraph,ξ( ) which we nearly show, instantiates using the inproper two perfectties of the sinusoids functional on theequation, CL. Itis that this the alter- cant figures (in the order of 15, 30 or 50, depending on the case) for floating point numbers. real and imaginary curves of the function regularly and alternately = −⁄ (cancel4! ) each other is a three-order polynomial in : out, thanks to () which nearly instantiates in two perfect sinusoids on the CL. It is this 24 17807 29 × 10079 43 × 11009087 13131 × 23027 = 7 − 7 + − + − 5 7×2! 7×5×3! 7×5×4! 7×5! 3458158927391587 7 × 47 × 2095343 × 148000093 + − +⋯ 3×5×7×9×11×6! 3×5×7×9×11×7!

168 168 () = ≈ + (1+)≈() 1 + ≈ ≈() 5 5

Information 2021, 12, x FOR PEER REVIEW 7 of 35

Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

2.5.5. Hadamard’s Product Formula Hadamard’s product formula contains in its expression, on the one hand, the non- trivial zeros and on the other hand, the functional Riemann equation. It therefore has a remarkable strength to characterize the zeta function and its deep relationship with neighborhoods of zeros. We know the formula: ′() 1 Γ′(1+½) 1 1 = ln2−½−1− −½ + + () −1 Γ(1+½) − (½) Moreover, we know that: ∑ =1+ ⁄ 2−ln2−½ln; = (1+½) (½) The formula becomes: ′() 1 1 = ln√ −½(1+½) − + () −1 − Information 2021, 12, x FOR PEER REVIEWInformation 2021, 12, x FOR PEER REVIEW 18 of 35 3 of 35 InformationUsing2021 a, CL12, 22point of ℒ and the zeros ⊂ℨ on the CL ℒ⊂, which assumes the 18 of 35 Information 2021, 12, x FOR PEER REVIEW 18 of 35 RH to be true, the infinite sum is written: ∑ =−∑ We finally obtain, taking into account the pairs of zeros on the CL (>0 and <0): alternation that generates a non-trivial zero in a restrictedTable 1. interval.Notations. On the CL, the calcu- ( ) nation that generates a non-trivial zero in a restricted interval. On the CL, the calculations ′ lations are extremelyalternationDefinition1 simplified: that generates 1 a non-trivial1 zero in a restricted interval.Notation On the CL, the calcu- = ln√ −½are extremely(1+½) simpliﬁed:+ − + () lationsAbsolute are̅ value extremely − simplified: + ∀ ∊ ℝ: || = > 0; − < 0 =½+; () =()(̂) =()()̅ =()() ⇒ |()| =1⇒() = 1 1 () = () +2( ) ( ) ( ˆ) ( ) ( ) ( )() | ∀( )| ∈ ℂ: = ( +) = iω= || ; =½+; = + i ; ζ( ) ==ξ−( )ζ(̂ )= = ξ( )ζ(̅ =) = ξ( )ζ( ⇒) ⇒ |ξ( =1⇒)| = 1 ⇒ξ(=) = e 2 Γ(̂⁄)2 Γ(2⁄) || = |+| = + ; , , , ∈ ℝ () = = ; ()() =1 This formula converges slowly on theComplex imaginaryΓ( number2⁄) component, ΓΓ((2̂⁄)⁄)2 becauseΓ (of2 ⁄)the terms in = (⁄) 2 > 0; Γ( ˆ /2) Γ( /2) . ξ(())== πi == πi ; ;(ξ)(()ξ =)(=1 ) = 1 (⁄) + < 0; Γ(2⁄) Γ(2⁄) By multiplying by () if Γ ( is not/2)( a⅝ zero:) Γ( (/2)) = =0: =()()(⁄) 2= ( >)| 0;( )| = −⁄ 2 < 0 When is very large, we can do the approximation: ≈ ≈ . ⅝ By multiplying by () if isIn not ℝ ,a the2 zero: integer ( )part = and( the)( fractional)() =2 part()| of( a)| real number : By multiplying( ) by ζ( ) if is not a zero: (ζ( )) = ξ(( ))ζ( )ζ( ) = ξ( )|ζ( )| Integer and fractional parts ⁄ ℑ The sum can then|( be)| ≠0 ⇒broken down() = into two sums= if ⇒⋘ () = ±: |()| ⇒ ∀ ∊=tanℝ: =[ ] + {}; [] ∊ℕ,0≤{} <1 |()| () ℜ() ℑ2() | (Complex)| number( ) in ζthe( )CS 2 In (ℂ,) =+ =ℜ| ( )| ⁄ () + ℑ(ζ)(;∈ℂ;,∊)) ω ℝ; its conjugate ̅ =− 1Information ≠0 ⇒ 2021, 12, x1 FOR= PEER REVIEW = 1iω ⇒ = ± iω⇒/2 =tan 5 of 35 |ζ( )=| 6= 0 ⇒ ξ( ) = +|()| = e ⇒ ζ( ) = ±|ζ( )|e ℜ⇒ () =2tan − − |ζ( )| −( ) =−( )|In( ℂ,)| ̂ =1−=1−− ;,∊R(ζ( )) 2 ℝ; ComplexIf number is zero, Hospital’s1− associated rule gives: with ̌=1− =̂̅ =1−+ If is zero, Hospital’s rule gives: ( )its conjugate=−( )| ( )| . We can then choose to obtain a good′( )approximation, but the 0convergence2 is ℜal- (0 ) 2 If n is zero, Hospital’s rule( gives:) (ζ ((n) ))=⁄ −ξ( n)|̂ ζsymmetric( n)| of with respect to (½, 0). |()| =0 ⇒() =− = ⇒ = ± ⇒− =tan ways slow. 0 2 2.5. Mathematical Tools Used 0 ζ |(′(n))| ′() ℑR(ζ(()())) ℜ2(ω)) | ( )| = ⇒ (|)(=)|−=0 ⇒Critical() =−strip= i(CS)ω ⇒ 0( )== ± ⇒ 0 ( () In ) = ± itheω/2 CS⇒ ( −): =ℜ⁄ ⇒− =+ ℑ = + ; 0≤≤1,>0; ≠1=tan . If wasζ always0 veryξ small compared0 to e, whichζ is neveri theζ case e since 0 tan |ζ ( n)| |′()|2.5.1. Stochastic Processes (ζ ( )) ℒ∶==½ℑ (2) ⁄=½+; >0 2 sweeps the entire range of ordinates,The wenon-trivialCritical could have line zeros (CL)an estimateare at the of breakingthe infinite point sum: of the On thetwo CL functions + : and . ⁄ is a random variable (an RV). The RVs are pairwise independent RVs. − . CS, except the CL \ ℒ∶=0≤≤1;≠½; ≠1iω/2 iω/2 +⁄ The non-trivial The zeros non-trivial are at the zeros breaking are at point the of breaking the two functionspoint of +thee twoand functions−e . and 1 1 ln⁄ − 1 is a Gaussian RV ~( , ). The RVs are pairwise independent. ⋘ ⇒ ≈ UsingUsing the the− power power. series series expansion expansion of of ΓΓ((2⁄)/2), ,we we serially( serially) = ∑ expand expand . In ξ (the()) CS,toto find ﬁnd the ω :: function is divergent. Zeta function: The two-layer hierarchical model ℳ is: 4 OneΓ considers(2⁄) in this CS, the analytical() continuation (). Using the()( power) series expansion of , we serially expand to find : |(1)|=1⇒1 + 2( +) = 2 ln 2= + −24ln ; iω = An⁄i( 4 αanamorphosis+−2β) {̃}; = −1 ⁄from(24 )ℂ; to ℂ: Êta|ξ (Dirichlet)| = 1 ⇒function:ξ( ) = e = e ; α = π/4 − 2π et ; θ =(−)1/=(24()();1−2). = () 4 |()| =1⇒6 () = = ; = ⁄ 4 −2{̃}; = −1 ⁄(24) ; Derivative of the function : ⊂ ℂ ⟶ ⊂ℂ () =−∑ log()⁄ 3i 9 139i 475 i 1 73i 131 iβ =1−3i − 9 + 139i + 475 +⋯1+ i − 1 − 73i + 131 +⋯ e = 11 − 1 8− 128+ 3i30729+ 32768139i+ ··· 4751 + 3 − 18 −i16201 +9720()(73i)=+∑··· 131−log() 8 k-order2 derivatives3 ( ) of the4 function 3 2 3 4 ≈ 1 +128=1− 2 + 2 ln3072 2− + ′′ 232768−ln+ + : +⋯1+(, 18) ⟼: 1620(,− ̃) = (−9720) ; ̃ = ln + (⁄); +⋯ =⁄ 2 24 8 128 3072 32768 In the CS,3 one18 uses the1620 analytical9720 continuation of these derivatives iβ ˘iθ i i 1 1 (6×7×24−5(6×7×24−)5i)i 7×247×242−5−5 1919 ×× 31× ×8707i 8707i 317317×531103 × 531103 e =e = 1 − − − − Two stochastic− processes {−}( and) = {()} are(̂) =adjoined()(1− by ) independent RVs on = =1− 24− − 2 3− 4 − 5 − 6 24 Functional2! (242!(i)24 equation) 5×3!1 5 ×of(3!24 the(24() 6×7×24−5) function5×4!5×4! ((2424)i )7×247×5!7×5!−5(24(24)19) ×35 3135× ×6! × 8707i( 6!24(24) )317 × 531103 Γ(̂⁄)2 ℒ associated with independent( ) Gaussian (RVs⁄) (on ℒ,) representing½ the nth zero. 13×5243712823i =13 ×=1−31 5243712823i×661×28660561− 31 ×27 661−×8197117457520809i × 28660561 27− × 8197117457520809i −=2 sin 2−Γ 1− = − 7 − 24 2!8 (24− ) 5×3!(24The)9 pair 5×4!, (24 forms) real7×5! RVs(24 on) the probability35 × 6! (24 space) Γ ((2Ω,⁄) , ℙ), measurable × − × − × × − 5 7!(24 ) 5×7!5 (8!24(2413) ×) 5243712823i5×8!25 11(24319!( ×)24 661) × 2866056125 × 11 ×27 9! ×(24 8197117457520809i())=()(̂) =()()̅ =()(̅ ); 81×5099×4517050132181 − − functions of (Ω, ) on− ℝ, ℬ(ℝ): − 81 × 5099+ ×··· 4517050132181 ( ⁄) 10 Functional5×7! equation(24) on the CL5×8! (24) 25 × 11 × 9! (24) Γ 2 11×5×10!(−24 ) +⋯ () = 11×5×10!81 ×( 509924) × 4517050132181 : ℝ,ℬ(ℝ) ⟶ ℝ,ℬ(ℝ) Γ(2⁄) − +⋯ WeWe can can therefore therefore11×5×10! extract (24 the) rotationrotation θ3 ===−1θ = −⁄1(/24(24)y )fromfromt= this this⁄( 2expansion: expansion:) = ⁄ 2 iβ iθ ⟼ ~( =, ); = ( ) e == e + + P3(iθ) ̃ =ln(⁄)= ⁄(2) ln(⁄)(2). The anamorphosis produces a stronger Anamorphosis: We can therefore ̃ extract the rotation ==−1⁄(24) from this expansion: ( ) elongation of the axis as increases. = + iθ ̃ ̃ ̃ ̃ β== arg e ++ (i(θ)))≈=−1≈ =θ = (−)1/⁄(=(2424)() ) ln = ()(⁄ 2) ln(2⁄) We3 define by the adjective “digressive” a series that exhibits a beginning simi- = + ()≈=−11 ½⁄(24) lar to a known = series, but which varies lnnoticeably with the subsequent terms. 3 isis a a three-order three-order polynomial polynomial in in iθ = −−i⁄/((4!4! ): ⁄ (2) 2 For example,⁄( the) cosine series is written: cos = 1 − + − +⋯+ 24 17807 is a 29three-order × 10079 polynomial43 × 11009087 in = − 4!13 :131 × 23027 ! ! ! = 7 − 7 + Digressive − series + − (⁄) 5 24 7×2! 178077×5×3! 29 × 100797×5×4!(−1) 43½ ×+ℴ 11009087 ( )7×5!. We will13 then131 ×note, 23027 in order to simplify the writing, a ()! = 7 − 7 + − + ln− 534581589273915877×2! 77×5×3! × 47 × 2095343(2) = ×7×5×4! 148000093 7×5! + − digressive cosine:(2) ⁄ (+⋯) =1− + − +⋯+(−1) + 3×5×7×9×11×6!34581589273915873×5×7×9×11×7!7 × 47 × 2095343 × 148000093 ! ! ! ()! + − ℴ(+⋯). 3×5×7×9×11×6!We can consider3×5×7×9×11×7! that the stochastic processes are Markov processes (at least for < 168 Complex zeros ℨ in the CS= 10 ): = + ; () =0 = 1 + 5 168 =½+ ; () =0 ( = | ,…, ) =( = | ); > ; ( = ≤ | ) =0 =41Zeros + ⊂ ℨ 36 on the CL ℒ⊂ The zeros such are that ordered in pairs, and 1− . . + 31 ×5 12 + (34 × 239 + 2 × 1643717 + 53 × 66797) >0 35 4 5 36 Calculations are carried out with 1 + (31= × 12 |+,…,(34) ×= 239( += 2 × 1643717|); the constraint+ 53 × 66797 is )>; ( =≤|) =0. + (68 ×Lambert 6135 × 107923 function × 9656115 + 23 × 4881469The main × branch 16227313 is defined) +⋯ by: () = ⇔= ;≥−1⁄ . 4511 1 Gamma function With this assumption, there is hereΓ( a) dependence= between two adjacent RVs. + (68 × 61 × 107923 × 965611 + 23 × 4881469 × 16227313 ) +⋯ We can deduce45 11 an approximate value of () on the CL: We canEuler’s deduce constant an approximate value ofEuler–Mascheroniξ( ) on the CL: constant: = lim− ln + ∑ ≈ 0.5772156649 We can deduce2.5.2. an approximate The Truncated value Gaussian of () Distribution on the CL: Density→ 168 ()() Taylor ≈ series ( 1+); 168 = −1⁄(24) () = ∑ (−) 5 168 The3 first two moments of an RV are the expectation! and the variance, the centered 3 ≈ i θ (1 + iθ); θ = −1/(24 ) ≈second5 ( 1+order )moment:; = −1 ⁄=(24[) ]; =[] =[ ] − ([ ]) . The RV which follows Bernoulli numbers 5 Coefficients of the power series of = ∑ ℬ ~(, ) ! 168 a( normal) distribution1683 is denoted by: () . ( ) ( ) 168θ3 =iω≈iα iθ +168 3 1+ ≈i(α+θ) 1 +168 θ ≈i(α+θ+ ) ≈i(α+θ) ξ( ) = e ≈ e e + i 5 θ (1 + iθ)168≈ e 1 + i 5 ≈ e 168 5 ≈ e ½ The Gaussian() probability density is: () = () ; standard law: () = () = ≈5 + (1+)≈5 1 + ≈ ≈√ 5 5 ½ √ A real RV is said to have density if there exists a positive and integrable function

on ℝ , called a density function, such that for all (, ) ∈ℝ: ℙ(≤≤) = ( ) . For an RV , of distribution function , of density , the conditioning at the interval [, ] gives:

ℙ() ()() ∊[, ]: ℙ(≤|≤≤) = = ;() = (). ℙ() ()() () The associated density is: () = . The truncated normal distribution that ()() we constrain to belong to the interval [, ] has the density:

(, , , , ) = ; − distribution function.

Information 2021, 12, x FOR PEER REVIEW 7 of 35

Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

2.5.5. Hadamard’s Product Formula Information 2021, 12, 22 Hadamard’s product formula contains in its expression, on the one hand,19 ofthe 35 non- trivial zeros and on the other hand, the functional Riemann equation. It therefore has a Information 2021, 12, x FOR PEER REVIEW remarkable strength to characterize the zeta function and its deep18 relationship of 35 with neighborhoods of zeros. We know the formula: Information 2021, 12, x FOR PEER REVIEW The functional equation on the CL does not represent a pure rotation, but this equation19 of 35

Information 2021, 12, x FOR PEER REVIEWallows to deﬁne an′( equivalence) relation, within1 the orderΓ′(1+½ of R3)◦ E. With this1 equivalence119 of 35

relation, the CL is a concatenation= ln2−½−1− of equivalence−½ class instantiations+ of unit length+ along alternation that generates a non-trivial() zero in a restricted−1 interval.Γ (On1+½ the CL,) the calcu-− the anamorphic axis t. e lations are extremelyThe functional simplified: equation on the CL does not represent a pure rotation, ( ½but )this equa- Moreover, we know that: ∑ =1+ ⁄ 2−ln2−½ln; = (1+½) The functional equation on the CL does not represent a pure rotation,(½ but) this equa- ( ) tion( ) allows( ) to( define) ( ) an equivalence( )() | relation,( )| within( ) the order of ℛ ∘ℰ. With this equiv- ∀=½+; 1, 2 ∈ L : 1≡=2 ⇔ et1̂ =≡Theet2 formula ⇔̅ ξ=( 1becomes:)= ξ( ⇒2 ) modulo =1⇒(R3 ◦ E) ⇔=aspect ζ( 1) ≡ aspect ζ( 2) alencetion relation,allows to the define CL is an a concatenationequivalence relation, of equivalence within the class order instantiations of ℛ ∘ℰ. ofWith unit this length equivalence relation, the CL is ã( concatenation) of equivalence class instantiations of unit length along theNow,Γ( anamorphiĉ⁄)2 we will beΓ (axis2 able⁄) ′. to determine the alternating zeros1 of the real and1 imaginary () =along the anamorphic= axis; (̃. = ln)(√)=1−½ (1+½) − + curves,Γ(2 by⁄) taking advantageΓ(2⁄)() of this evaluation via an approximate−1 calculation− of the ∀, ∈ℒ: ≡ ⇔ {̃ } ≡ {̃} ⇔() =() (ℛ ∘ℰ) ⇔ () ≡ () ∀ , ∈ℒ: ≡derivative.⇔ {̃ } ≡ {̃ } ⇔( ) =( ) (ℛ ∘ℰ) ⇔ ( ) ≡ ( ) Using a CL point of ℒ and the zeros ⊂ℨ on the CL ℒ⊂ , which assumes the By multiplyingNow, by we ( will) if be able is not to dete a zero:rmine the() alternating = () zeros()( of) the= real()| and()| imaginary curves, RH to be true, the infinite sum is written: ∑ =−∑ by3.6.3. taking TheNow, advantage Zeros we will of of thebe this able Real evaluation to and dete Imaginaryrmine via anthe approximate alternating Parts of the zeros calculationζ Function of the real of the and derivative. imaginary curves, () ℑ() |()| ≠0 ⇒() = by takingIn this=We advantage paragraph,finally ⇒ ( obtain, of) = ± this we |evaluationtaking( perform)| into⁄ via ⇒acco the anunt calculationsapproximate the=tan pairs of calculation assuming zeros on the of the theCL approximation: derivative.(>0 and <0): |()| ℜ() 2 3.6.3.ω = Theθ1 + Zerosθ2 + θof3 +theθ′ 4Real(.) and Imaginary Parts of the1 Function 1 1 3.6.3. The Zeros of the Real and Imaginary Parts of the Function InOn this the paragraph, CL, we can we calculate= ln perform√ the−½ the derivative calculations(1+½ of) +ξ( assuming)− at constant the xapproximation:, as a+ function of=y: If is zero, Hospital’s rule gives:() () =−()| ()|̅ − + In this paragraph, we perform the calculations assuming the approximation: = + + +. 1 ++ + 0 = () +2(2 ) ′ω(On=) πthe/4 CL,−2 πweet .can +( )calculateθ ⇒ ω =( the)− derivativeln⁄t + 1/ ℜ4! of () = −at lnconstant t− θ/ ; ,θ as= aθ function3 = −1/ (of24 : ) |()| =0 ⇒() =− = ⇒ = ± ⇒− =tan − |′( )|On the CL, we can calculate the derivative of () at 2constant , as a function of : ℑ () = ⁄ 4−2{̃} + ⇒This =−ln+1formula converges⁄(4! 0) slowly=−ln− on the⁄ imaginary; = =−1 component,⁄(24) because of the terms in 0 iω 0 iω iω 2 = ⁄ 4−2{̃} + ⇒ =−ln+1ξ ( ) = e⁄(4! =)iω=−ln−e = ie ⁄ −; =ln t + 1/ =−14! ⁄(24⁄ ) The non-trivial zeros. are at the breaking point of the two functions + and −⁄ . ′() = () =′ =(−ln+1⁄)(4! ) (⅝) When is very large,ζ we() can do the approximation: ≈ ⅝ ≈ . The′( values) = ( of ) =′for which= (−ln+1cancel each⁄)(4! other ) out for the real part (resp. the Using the power series expansion of Γ(2⁄), we serially expand () to find : imaginary part), correspond to the values for which ξ( ) = −1 (resp. ξ( ) = 1). They co- The valuesThe sumof canfor thenwhich be ()broken cancel down each into other two outsums for if the ⋘ real part: (resp. the im- The values() of for which () 5 cancel each other1 out for the real part (resp. the im- |()| =1⇒(aginaryincide) = part), with= thecorrespond values ; = close to⁄ 4 the−2 to values{et̃}; = for =8 (resp. −1which⁄(24 et()=); =−18 ), that (resp. is to say() the=1). values They αcoin-= π, aginary= part), correspond to thẽ values1 for which̃ (1) =−1 (resp. ()1=1). They coin- cide(resp. withα the0). values close to {} =⅝ (resp.= {} =⅛), that+ is to say the values =, { ̃} { ̃} 3i 9 139i(resp.cide We=0 with assume475). the values in the close calculations: ito −1=⅝ω =(resp.73iα + θ==⅛−π131/),4 that− 2π is{ to/ (say2π) theln−( values/(2πe ))=} − , =1− − + + +⋯1+ − − + +⋯ (resp.( )=0). 8 128 30721/ We24 32768assume. We can in then the choose calculations:3 18 to obtain = +=1620 a good 9720approximation,⁄ 4−2{⁄( 2but) lnthe( convergence⁄)}(2) − is al- We assume in the calculations: = +=⁄ 4−2{⁄(2) ln(⁄)}(2) − 1⁄(24ways). slow. i 1 (16×7×24−5⁄(24). )i 7×24 −5 19 × 31 × 8707i 317 × 531103 = =1− − − If( ) was(− )always( ) ( very) small− compared to− ( , which) is( )never the case since 24 2! (24) 5×3!(24 =) 5×4! ̅ ; (24 =) +; 7×5! +=(24) 35 − × 6! (24; ) =+; sweeps (the) =entire() range()̅ ; of( ordinates,) = +; we coul+=d have an( estimate−); (of )the=+; infinite sum: 13 × 5243712823i 31 × 661 × 28660561 27 × 8197117457520809i − − − Case where {} =⅛;5×7! (24) 5×8!(24) 125 × 111 × 9! (24ln) − 1 ⋘ ⇒ ≈ {81} ×=⅛; 5099 × 4517050132181 Case where ( ) ⁄(!)4 − =0 ; = +⋯= ≈ = ≈1− − +⋯( ) 11×5×10!=0 (24; )() = = ≈ = 1⁄(!1) ≈1− + 2 + 2 ln− 2 + +⋯−24ln +=ℯ =(− ) 4 ⁄( ) 6 1 − ℯ+1+ℯWe can therefore=0⇒sin extract (the2+=ℯ⁄) rotation= cos ((2−⁄)==−1; ) ⁄ =tan24(2 ⁄)from≈−tan this (expansion:1⁄)(48) = + () ( ⁄) ( ⁄) ⁄ ( ⁄) ( ⁄)( ) 1 − ℯ +1+ℯ =0⇒sin =1−; =−1 21 = cos 2; =tan 2≈−tan 1 48 ( ) =1−; =− ≈ 1 + 2 + 2 ln 2 + ′′ 2 −ln + =1 ⇒⁄ =−⁄ ; −2+24 =0 ⇒≈ ⁄2 ⇒⁄ = /2 = + ()≈=−1 ⁄(24 ) It follows then that: ()≈1−+ =1 ⇒⁄2 +=1−1⁄ =−⁄⁄(1152; −2+) −⁄(24=0 ⇒≈) ; ⁄ =⁄2 ⁄ 2⇒≈−1⁄ ⁄= /2(48) It follows then that: () ≈1−⁄2 +=1−1⁄( =1152 −⁄(4!) −) ⁄(24) ; ⁄ =⁄ 2≈−1⁄(48) To improve the estimation is a three-order of , such polynomial that () in=½+ ,=+=: , we must take into account the deriva- To improve the estimation of , such that () =½+,=+= , we must take into account the derivative of 24. Hence, we can calculate17807 the29 difference × 10079 Δ with43 ×the 11009087 pivot value 13,131: × 23027 tive = of . Hence,7 − 7 we + can calculate − the difference −1 Δ+⁄(24 with) the pivot value − ,1 : 5 7×2! 7×5×3! 7×5×4! 7×5! Δ = = −1⁄(24) = 24 ln − 1 3458158927391587Δ = −ln+1= 7 × 47⁄( ×4! 2095343 ) = × 24 1480000932 ln − + −−ln+1⁄(4! ) 2 +⋯ The approximate value3×5×7×9×11×6! of for which () has a null 3×5×7×9×11×7!imaginary value is therefore: The approximate value of for which () has a null imaginary value is therefore: ½ + ,= + 0⇒, =, +Δ 168 =−2+⅛; ½ + ,==2 + 0⇒⁄ ⁄, ;=,=+Δ ⁄(2) = 1 + , , , , , , , =−2+⅛; , =2,⁄,⁄ ;, =,⁄(2) 5 ≈, +1⁄ 4!, ln , −1⁄ , 4 36 ≈, +1⁄ 4!, ln , −1⁄ , ∀ ∈+ ℕ: (31) =½+ × 12 +,(34= × 239 ⇒ +( 2) =1 ; × 1643717, ≈ ,+ 53+1 × 66797⁄ 4!,)ln , −1⁄ , ∀ ∈35 ℕ: () =½+ 5,= ⇒() =1 ;, ≈, +1⁄ 4!, ln , −1⁄ , 1 Case where {} =⅝;+ (68 × 61 × 107923 × 965611 + 23 × 4881469 × 16227313) +⋯ 4511 Case where {} =⅝; () ⁄(!) = ; () = = =− ≈− = − ≈−1− − +⋯ = We ;can ( deduce) = an= approximate() =− value≈− of = −() on⁄ the(! )CL:≈−1− − +⋯ +=(−) =−ℯ(−) 168 +=(−) =−ℯ(−) 1 + ℯ +1−ℯ ≈ = 0 ⇒(1+ cos)(; 2⁄) = −1=sin⁄(24(2)⁄) ; ⁄ =tan(2⁄)≈tan (1⁄)(48) 1 + ℯ+1−ℯ 5 = 0 ⇒ cos(2⁄)=sin(2⁄); ⁄ =tan(2⁄)≈tan (1⁄)(48) =−1+; = =−1+; = ⁄ =⁄ ; −2+ =0 ⇒= ⁄2 ⇒⁄ = /2 168 168 It follows() = then that:≈ () ≈−1++ ⁄(⁄1+2 =+=−1+1⁄) ≈; −2+()⁄1(1152 +=0 ⇒= ) +≈⁄(24⁄2 )⇒; ⁄⁄ == /2≈⁄ 2(≈1 )⁄ (48) It follows then that: () ≈−1+5 ⁄2 +=−1+1⁄(11525 ) +⁄(24) ; ⁄ =⁄ 2≈1⁄(48) , =−2+⅝; , =2,⁄,⁄ ;, =,⁄(2) , =−2+⅝; , =2,⁄,⁄ ;, =,⁄(2) ∀ ∈ ℕ: () =½+,= ⇒() =−1 ;, ≈, +1⁄ 4!, ln , −1⁄ , ∀ ∈ ℕ: () =½+,= ⇒() =−1 ;, ≈, +1⁄ 4!, ln , −1⁄ ,

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Case where =⅝; = ; () = =() =− ≈− = −⁄(!) ≈−1− − +⋯ +=(−) =−ℯ(−) 1 + ℯ+1−ℯ = 0 ⇒ cos(2⁄)=sin(2⁄); ⁄ =tan(2⁄)≈tan (1⁄)(48) =−1+; = ⁄ =⁄ ; −2+ =0 ⇒=⁄2 ⇒⁄ = /2 It follows then that: () ≈−1+⁄2 +=−1+1⁄(1152) +⁄(24) ; ⁄ =⁄ 2≈1⁄(48) Figure ,1. Comparison=−2+⅝; between , the=2 Lambert,⁄ and, Logarithm⁄ ;, functions:=,⁄ (Lambert’s2) function compresses high values more ∀ ∈ ℕ: (than) =½+ the Logarithm,= function. ⇒ () =−1 ;, ≈, +1⁄ 4!, ln , −1⁄ ,

Case where =⅜; 2.5.5. Hadamard’s Product Formula Hadamard’s product formula contains1 in its expression, on the one hand, the non- =−2; () = =(⁄ ) =− ≈− = −⁄(!) ≈−− + +⋯ trivial zeros and on the other hand, the24 func1152tional Riemann equation. It therefore has a +=remarkable(− strength) =−ℯ to (characterize−) the zeta function and its deep relationship with neigh- It follows that: () ≈−(1−⁄)2 =−(1−borhoods⁄)2 =−1of zeros.⁄(24 We) −+know the⁄(1152 formula:) ; ≈− =−2+⅜; =2 ⁄ ⁄ ; = ⁄(2) , , ,′() , , , 1 Γ′(1+½) 1 1 ∀ ∈ ℕ: () =½+ = (1 − ) ⇒ () =− ;= ln2−½−1−≈ +1⁄ 4! −½ln −1⁄ + + , () , , −1, ,Γ(1+½), − (½) Case where =⅞; Moreover, we know that: ∑ =1+ ⁄ 2−ln2−½ln; = (1+½) 1 − (½) = 2; () = =(⁄ ) =The formula≈ = becomes: ⁄(!) ≈+ + +⋯; 24 1152 ′() 1 1 += (−) =ℯ (−= ln) −½(1+½) − + √ It follows that: () ≈+(1− ⁄)2 =−+(1− ⁄)2 =1⁄(24()+−) ⁄(1152 ) ; ≈ −1 − , =−2+⅞; , =2Using, a ⁄CL point,⁄ ;of, ℒ =and, the⁄ zeros(2) ⊂ℨ on the CL ℒ⊂, which assumes the ∀ ∈ ℕ: () =½+,=(1 + ) ⇒ () = ;, ≈, +1⁄ 4!, ln , −1⁄ , RH to be true, the infinite sum is written: ∑ =−∑ We finally obtain, taking into account the pairs of zeros on the CL (>0 and <0): 3.7. The Consecutive Zeros on the CL 3.7. The Consecutive Zeros on the CL ( ) Figure 15a shows the rather′ asymmetrical histogram of the1 raw distances 1between 1 Figure 15a shows the rather asymmetrical= ln histogram√ −½ of( the1+½ raw) distances+ − between two + two consecutive zeros from . The( )tail of the histogram has not been̅ represented − for val-+ consecutive zeros from D. The tail of the histogram has not been represented for values ues greater than 1.5: these are some large differences of the first zeros (1 < 50,000). Fig- greater than 1.5: these are some large differences of the= ﬁrst( zeros) +2 ( < 50, 000). Figure 15b ure 15b represents the scatterplot of these distances along the ordinate. − We observe represents the scatterplot of these distances along the ordinate. We observe that these that these distances, on average, tend to decrease with , and we understand from this distances, on average, tend toThis decrease formula with converges, and we slowly understand on the fromimaginary this ﬁgure component, that the because of the terms in figure that the raw axis is not the most appropriate variable to study the statistics of raw axis is not the most appropriate. variable to study the statistics of gaps between zeros. gaps between zeros. It is preferable to analyze the population of the gaps between zeros,(⅝ ) It is preferable to analyze theWhen population is very of the large, gaps we between can do zeros,the approximation: after anamorphosis, ≈ ⅝ ≈ . after anamorphosis, along the ̃ axis. along the et axis. Figure 16a,b showThe the sum graphs, can thenafter bethe broken transformation down into of two the sums axis if into⋘ the :̃ axis. We greatly reduced the influence of the bias of 1the use of the variable1 . There is only1 a slight asymmetry, which can be interpreted as follows. = We selected the + zeros in the order − − − of appearance on the axis, so that the difference between and is always positive, as is the differenceWe ∆cañ =theñ choose−̃. It goes to obtain without a goodsaying approximation, that there are butpossible the convergence is al- underlying permutationsways slow. between the indices and +1. If there was a way to identify these permutations of indicesIf was of consecutive always very zeros, small especially compared when to the ,zeros which are is relatively never the case since close, the gaps betweensweeps some the entire adjacent range zeros of ordinates, could be wenegative, could haveand inan this estimate case, ofthe the left infinite sum: part of the histogram would have a branch which1 would1 overflowln − towards 1 negative values, still with mean 1. We⋘ would ⇒then obtain ≈a much more symmetrical and Gaussian 4 histogram. 1 1 + 2 + 2 ln 2 + () −24ln = 4 6

1 1 Figure 15. (a) Histogram of the gaps between two consecutive zeros, calculated on the raw y axis, y > 500, 000. Indeed, ≈ 1 + 2 + 2 ln 2 + ′′(2)−ln the ﬁrst ∆y is an outlier at 6.887; (b) the scatterplot of the gaps between two zeros24 as a function of the y ordinate.

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Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

1 1 ≈ 1 + 2 + 2 ln 2 + ′′(2)−ln 24

Figure 16. (a) Histogram of the gaps between two consecutive zeros calculated on the anamorphic et axis; and (b) scatterplot of the gaps between two zeros as a function of the anamorphic ordinate et.

The model expresses it clearly. The difference between two independent mixed RVs Ten+1 and Ten is a new Gaussian RV ∆Ten = N (µe∆, eσ∆n) of parameters µe∆ = µen+1 − µen = (n + 1) − n = 1; eσ∆n = eσn. We can calculate the average distance between two consecutive zeros ∆yn = yn+1 − yn:

2πt5,n+1 2πt5,n 2π 5 ∀n : ∆µn = µn+1 − µn = − ≈ ; t5,n = n − 2 + . W(t5,n+1/e) W(t5,n/e) W(t5,n/e) 8

2π 2π y ∀n : ∆y ≈ ≈ ; ∆t × ln(t /e) ≈ 1; t = n n 5 ln(y /(2πe)) n n n 2π W n − 2 + 8 /e n We can get, for N < 107, an estimation of the drift of extreme values of ∆y as a function of y: 2π/200 0.01π 2π × 2 × 4.5σ 6π ∀ : = < ∆y < = ln(y/2πe) ln(y/2πe) ln(y/2πe) ln(y/2πe)

The passage from y to et makes it possible to better compare the differences in the list of non-trivial zeros from D: some adjacent zeros are very close or far apart from each other (Figure 17). Information 2021, 12, 22 22 of 35

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alternation that generates a non-trivial zero in a restricted interval. On the CL, the calculations are extremely simplified:

Figure 17. Two very close zeros (a) and two very distant zeros (b). On the CL, the curves ζ(=½+; ) have conﬁgurations() = where()(̂) =()()̅ =()() ⇒ |()| =1⇒() = the undulations are very weak, in the vicinity of a very large fold. Γ(̂⁄)2 Γ(2⁄) () = = ; ()() =1 This is the case, without being exhaustive, in Table4, for the following pairs of zeros.Γ(2⁄) Γ(2⁄) Table 4. (a) List of near consecutive zeros; and (b) the list of distant consecutive zeros. By multiplying by () if is not a zero: () = ()()() =()|()| (a) The Smallest Gaps between Zeros (b) The Biggest Gaps between Zeros () ℑ() |()| ≠0 ⇒() = = ⇒ () = ±|()|⁄ ⇒ =tan ~ ~ |()| ℜ() 2 n ∆tn yn ∆yn n ∆tn yn ∆yn 1,115,578 0.0054468 663,318.508 0.002959 3,206,674 3.3637376 1,746,643.719If is zero, 1.686 Hospital’s rule gives: () =−()| ()| 5,042,995 0.0061228 2,650,984.611 0.002970 1,493,008 3.3034734 865,897.915 1.754 3,637,897 0.0065627 1,961,773.993 0.003259 1,793,281 3.2257130 1,024,176.503′ 1.689( ) ℜ() |()| =0 ⇒() =− = ⇒ () = ±()⁄ ⇒− =tan 3,271,858 0.0083140 1,779,292.804 0.004161 1,663,325 3.2221292 955,946.684 1.697 |′()| ℑ () 2 3,966,813 0.0083818 2,124,605.420 0.004137 2,865,881 3.2146956 1,575,130.462 1.625 420,891 0.0096951 273,193.663 0.005704 880,694 3.2142617 534,572.843The non-trivial 1.780 zeros are at the breaking point of the two functions +⁄ and 4,156,678 0.0098793 2,218,149.333 0.004859 5,640,489 3.2016993 2,939,651.982−⁄ . 1.541 4,980,819 0.0100586 2,620,810.352 0.004884 3,867,228 3.1954605 2,075,412.008Using the 1.580 power series expansion of Γ(2⁄), we serially expand () to find : 4,398,867 0.0111705 2,337,027.805 0.005472 5,491,901 3.1935420 2,868,076.402 1.540 1,205,484 0.0117584 712,004.001 0.006348 4,847,419 3.1773669 2,555,978.540|()| =1⇒() 1.546= =() ; = ⁄ 4 −2{̃}; = −1⁄(24) ;

3i 9 139i 475 i 1 73i 131 =1− − + + +⋯1+ − − + +⋯ 3.8. The Multiplicity of Zeros 8 128 3072 32768 3 18 1620 9720 It follows from the numerical value of the standard deviation of the Gaussian RVs i 1 (6×7×24−5)i 7×24 −5 19 × 31 × 8707i 317 × 531103 that the multiplicity of zeros is achievable thanks to the constant mixing of the close RVs = =1− − − − − − and that the possibility of this conﬁguration increases with the24 value2! ( of24n.) It could5×3! be at(24) 5×4!(24) 7×5!(24) 35 × 6! (24) 13 × 5243712823i 31 × 661 × 28660561 27 × 8197117457520809i the maximum of order 3 when n < 107: it would then be necessary that the (n − 1)th zero − − − arrives late, the nth arrives in advance, and that in addition, the (n +5×7!1)th( also24) occurs very5×8!(24) 25 × 11 × 9! (24) 81 × 5099 × 4517050132181 − +⋯ 11×5×10!(24)

We can therefore extract the rotation ==−1⁄(24) from this expansion: = + ()

= + ()≈=−1⁄(24)

168 168 () = ≈ + (1+)≈() 1 + ≈ ≈() 5 5

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Information 2021, 12, 22 Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses 23high of 35 values more than the Logarithm function.

2.5.5. Hadamard’s Product Formula early. This scenarioHadamard’s never appears product in the firstformula few millioncontains zeros in its and expression, its probability on the is one also hand, the non- very low: trivial zeros and on the other hand, the functional Riemann equation. It therefore has a 1 1 remarkableProb{etn + strength< {Ten− to1} characterize, {Ten}, {Ten+ 1the} < zetaetn + function+ ∆et }and= pits deep relationship with neigh- 8 8 borhoods of zeros. We know the formula: p = p p p ∆t3 = 0.3962 × 0.645 × 10−4(1/2π ln(y/2π)∆y)3 ≈ 10−5(1/2π ln(y/2π)∆y)3; 1 2 3 e ′() 1 Γ′(1+½) 1 1 = ln2−½−1−7 −3 −½−15 + + y =(10) ; ∆y = 10 : p ≈ 1.2−1× 10 Γ.(1+½) − The possibility of a double root is more likely, although it is known that, among(½) Moreover, we know that: ∑ =1+ ⁄ 2−ln2−½ln; = (1+½) the trillion zeros calculated, no double roots have yet been identified. The probability(½) is Information 2021, 12p, x= FORp pPEER∆t2 REVIEW≈ 8 × The10− 9formula, for y = becomes:107; ∆y = 10−4, knowing that the nearest zeros on D 18 of 35 1 2 e y ≈ ( ) nth = are distant from ∆ 0.003. We know′ [15] that between the 4, 088,1 664, 936, 2181and Information 2021, 12, x FOR PEER REVIEW ≈ = ln√ −½ (1+½) −3 of 35 += −9 the (n + 1)th, the gap is ∆y 0.00001710() . Notwithstanding, the probability−1 (p 10−) for this rare event is “quite large”. alternation that generates a non-trivialℒ zero in⊂ℨ a restricted interval.ℒ⊂ On the CL, the calcu- Hadamard’s productUsing a formula, CL point dynamically of and interpretedthe zeros via geometric on the CL analysis, , pro-which assumes the lations are extremely simplified: Table 1. Notations.RH to be true, the infinite sum is written: ∑ =−∑ vides a better understanding of the possibility of the multiple root phenomenon. We know 0 0 iω/2 that if =½+; p is a simpleWe( root) finally=ζ (p )obtain,=(̂)0,=ζ taking(p) (=into)̅ ±= iacco ζ(unt)p( thee) ⇒ pairs|is( not )|of =1⇒zeros null. Bothon( the) members= CL (>0 and <0): Definition Notation of Hadamard’s formula equilibrate by the term 1/ p − p = ∞, present in the second Absolute value ∀ ∊ ℝ: || = ′() > 0; − < 0 1 1 1 member: Γ(̂⁄)2 Γ(2(⁄) ) = ln √ −½ () 1+½ +− + ∀ ∈ ℂ: = ( +) =( =) = |=| ; ; ()(̅ ) =1 − + Γ(2⁄) Γ(2⁄) 0 ± 0 iω/2 ∞ 1 |ζ| =p|+|i =ζ p + e ; , , , = ∈ℝ( 1) +2 1 = = f p + + 2i p Complex number = By multiplying(⁄) 2 by () if > 0; is not a zero: ∑()−2= (2)()() =()|()| ζ p 0 0 n=1,n6=p n − p = (⁄) + < 0; This formula() converges slowly on the imaginary component,ℑ() because of the terms in =0: =⁄ 2 > 0; = −⁄ 2 < 0 ⁄ We|( study,)| ≠0 ⇒ below,. ( the) = configuration = of ⇒ the ζ(function) = ±|( when)| two⇒ zeros are very=tan close. |()| ℜ() 2 InWe ℝ know, the integer that when part theand real the partfractional (resp. part the imaginaryof a real number part) of: the derivative of ζis(⅝ null,) Integer and fractional parts When is very large, we can do the approximation: ≈ ⅝ ≈ . ∀ ∊ ℝ: =[] + {}; [] ∊ℕ,0≤{} <1 the imaginary part (resp.If theis zero, real Hospital’s part) of the ruleζ function gives: is located( ) =− on a( local )| extremum:( )| Complex number in the CS In ℂ, =+ =ℜ0 () + ℑ();∈ℂ;,∊ℝ; its conjugate ̅ =− 1 R(ζ ( )) = 0 ⇒The sum(ζ( can)) on then a local be broken extremum down for into the twoζ curve sums on if the ⋘x = 2:axis. 0 1 (ζ ( ))In =ℂ,0 ⇒̂ =1−=1−− ;,∊R(ζ( ′))(on) a local extremumℝ; for the ζ curve on theℜ x =()axis. 1− |()| =0 ⇒() =− = ⇒1 () = ±()1⁄ ⇒− 2 =tan1 Complex number associated with ̅ = + We observeits conjugate on the graphš =1−|′( (Figure)|=̂ =1−+ 18), that for. a double root to exist,ℑ it( would) be 2 − − − = 1 necessary that̂ symmetricζ (both real of and with imaginary respect to parts) (½, 0 be). tangent to the CL x 2 . We would ⁄ TheWe cannon-trivial then choose zeros are to atobtain the breakinga good approximation, point of the two but functionsthe convergence + is and al- Critical strip (CS) Inthen the beCS dealing: =ℜ with(⁄) + ℑ an extremum() = + ; 0≤≤1,>0; ≠1 (local maximum or minimum). of the function for the two curves. We−ways would slow.. therefore have in this case ζ( ) = ζ0( ) = 0. If there is such a Critical line (CL) On the CL ℒ∶==½: =½+; >0. Γ(2⁄) () double root, the behaviorUsingIf was ofthe the poweralways two membersseries very expansionsmall of Hadamard’scompared of to formula , we , seriallywhich is changed. isexpand never In the caseto find since : CS, except the CL \ ℒ∶=0≤≤1;≠½; ≠1 sweeps the entire range of() ordinates, we could have an estimate of the infinite sum: rest of this paragraph,|()|=1⇒ it is( no) = longer= assumed ; that = the⁄ 4 zeros−2{ arẽ}; all = on −1 the⁄(24 CL.) We; recall () = ∑ . In the CS, the function is divergent. Zeta function: that if s ∈ Z is a zero of the CS, then sˆ = 1 − s, s, 1 − s are also zeros. Now, we examine the 1 1 (ln) − 1 geometricOne considers configuration in this⋘ CS, of twothe analytical⇒ different zeroscontinuation≈ belonging to. two neighboring quartets if 3i 9 139i 475 i 1 73i 131 4 Êta Dirichlet function: =1− − (+) =()(1−2+ ). +⋯1+ − − + +⋯ they are both8 in128 the CS, in3072 two pairs32768 if they are both on the3 CL,18 or in a quartet1620() and9720 a duet if () =−∑ log()⁄1 1 + 2 + 2 ln 2 + −24ln Derivative of the function they are one in the CS and the other on= the CL. 4 6 i ()(1) = ∑(6×7×24−5−log() )i 7×24 −5 19 × 31 × 8707i 317 × 531103 k-order derivatives of the function = =1− − − − − − In the CS, one24 uses2! the(24 analytical) 15×3! continuation1(24) of these5×4! derivatives(24) 7×5!(24) 35 × 6! (24) ≈ 1 + 2 + 2 ln 2 + ′′(2)−ln 13 ×( 5243712823i) =()(̂)=31( ×) 661(1− × 28660561) 27 × 8197117457520809i − − 24 − Functional equation of the function 5×7!(24) 5×8!(24Γ)( ̂⁄)2 25 × 11 × 9! (24) () =2 sin(⁄) 2 Γ(1−) =½ 81 × 5099 × 4517050132181 Γ(2⁄) − +⋯ () =11×5×10!()(̂) =(24())()̅ =()(̅ ); ( ⁄) Functional equation on the CL Γ 2 We (can) = therefore extract the rotation ==−1⁄(24) from this expansion: Γ(2⁄) = + () t=⁄(2) = ⁄ 2 ̃ =ln(⁄)=⁄(2) ln(⁄)(2) . The anamorphosis= produces + a( stronger)≈=−1 ⁄(24) Anamorphosis: ̃ elongation of the axis as increases. Figure 18. Plot for two near zeros: is aζ three-orderfunction (solid polynomial lines) and ζ in0 derivative = −⁄ (dotted(4! ): lines). We define by the adjective “digressive” a series that exhibits a beginning similar to a known24 series, but which17807 varies noticeably29 × 10079 with the subsequent43 × 11009087 terms.+ 13131 × 23027 = It is assumed7 − 7 that + the points −Rm and Rm+1of+ indices m and m 1 are− roots with very For example,close coordinates5 the cosinesm seriesand7×2! sism +written:1. We7×5×3! choose cos = the 1 − point+O7×5×4!with− +⋯+ coordinates s in the7×5! middle 3458158927391587 7 ×! 47 ×! 2095343! × 148000093 Digressive series (−1) +ℴ()+. We will then note, in order to− simplify the writing, a +⋯ ()! 3×5×7×9×11×6! 3×5×7×9×11×7! digressive cosine: () =1− + − +⋯+(−1) + 168 ! ! ! ()! = 1 + ℴ(). 5 Complex zeros ℨ in the CS 4 = + ; 36() =0 + 31 × 12 + (34 × 239 + 2 × 1643717 + 53 × 66797) 35 =½+ ; 5 () =0 Zeros ⊂ℨ on the CL ℒ⊂ The zeros1 are ordered in pairs, and 1−. + (68 × 61 × 107923 × 965611 + 23 × 4881469 × 16227313) +⋯ Calculations4511 are carried out with >0 Lambert function The main branch is definedWe can by: deduce ( )an= ⇔= approximate value;≥−1 of ⁄( . ) on the CL: ( ) Gamma function Γ = 168 ≈ (1+ ); = −1⁄(24) Euler’s constant Euler–Mascheroni constant: = lim−5 ln + ∑ ≈ 0.5772156649 → () () Taylor series () = ∑ (−) 168 168 ! () () () = ≈ + (1+)≈ 1 + ≈ ≈ Bernoulli numbers Coefficients of the power5 series of = ∑ ℬ 5 !

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1 of the segment RmRm+1: s = 2 (sm + sm+1). The Hadamard formula applied to point O becomes: ζ0(s) √ 1 1 1 1 1 1 = − 0 + − + + + ∞ ln π ψ 1 s ∑n=1;n6=m, m+1 ; ζ(s) 2 2 s − 1 s − sm s − sm+1 s − sn Γ0 1 + 1 s 0 1 2 ψ 1 + s = 2 1 Γ 1 + 2 s

O being the middle, we obtain: 1 + 1 = 0. The second member becomes a function s−sm s−sm+1 of s which takes ﬁnite real and imaginary values of C:

√ 1 1 1 1 − 0 + − + ∞ = ( ) ∈ \{ } ln π ψ 1 s ∑n=1;n6=m, m+1 f s C ∞ 2 2 s − 1 s − sn

The ζ functionInformation is holomorphic; 2021, 12, x FOR we PEER can REVIEW write at point O: 7 of 35

∆s (∆s)2 ζ(s) = ζ(s ) + ζ0(s ) + ζ00 (s ) + ··· ; ∆s = s − s m 1! m 2! m m The derived function is holomorphic; we can write at point O:

∆s (∆s)2 ζ0(s) = ζ0(s ) + ζ00 (s ) + ζ000 (s ) + ··· m 1! m 2! m 0 ∼ In the configuration of close zeros, ζ(sm) = 0 and ζ (sm) = ε = ε1 + iε2. It follows that: ! ζ0(s) ε + iε + ∆s × ζ00 (s ) + ··· 1 ε + iε + ∆s × ζ00 (s ) + ··· = 1 2 m = 1 2 m ζ(s) 1 2 00 ∆s + + 1 × 00 ( ) + ··· ∆s × (ε1 + iε2) + 2 (∆s) × ζ (sm) + ··· ε1 iε2 2 ∆s ζ sm Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more 0 1 00 000 ! ζ (s) 1 than the Logarithm∆s(ζ function.(sm) + ∆s × ζ (sm)) + ··· 2 ≈ 1 + 2 ≈ 1 00 1 2 000 ζ(s) ∆s ε + iε + ∆s × ζ (sm) + (∆s) ζ (sm) + ··· ∆s 1 2 2 2.5.5. Hadamard’s6 Product Formula When s and s converge, the midpoint s also converges and we obtain: Information 2021, 12, x FOR PEER REVIEW m m+1 Hadamard’s product formula3 containsof 35 in its expression, on the one hand, the non-

ζ0trivial(s) zeros2 and on the other hand, the functional Riemann equation. It therefore has a ∆s → 0; ε1 → 0; ε2 → 0; ⇒ remarkable≈ → strength∞ (∈ C to∪ characterize{∞}) the zeta function and its deep relationship with neigh- ζ(s) ∆s Table 1. Notations. borhoods of zeros. We know the formula: 00 It goes without saying that if ζ (sm) = 0, we extend′( the) expansion until we1 find Γ′(1+½) 1 1 Definition (k) Notation a non-zero derivative ζ (sm) 6= 0, which is always feasible= ln2−½−1− since the ζ function is not−½ + + () −1 Γ(1+½) − Absolute value uniformly null. In conclusion,∀ ∊theℝ: first|| = member > takes 0; − an infinite < 0 complex value, while the ∀ ∈ ℂ: = + = = || (); (½) second member is finite, which justifies that the doubleMoreover, root we configuration know that: is∑ impossible.=1+ ⁄ 2−ln2−½ln; = (1+½) (½) The holomorphic propertyInformation| that| =2021| is+, 12 included, x FOR| = PEER in Taylor’s+ REVIEW ; , formula , , ∈ ℝ accounts for the neighbor- 18 of 35 The formula becomes: Complex number hood very close to zero. It is essentially = ( the⁄) analysis 2 of the local > 0;extrema of the ζ function in relation to the behavior of the = derivative,(⁄) i.e., the observation + of 0; = −⁄ 2 < 0( ) −1 − of the real and imaginary parts that triggers the justification.alternation that generates a non-trivial zero in a restricted interval. On the CL, the calcu- In ℝ, the integer part and the fractional part of a real number : Integer and fractional parts lationsUsing a are CL extremely point of simplified: ℒ and the zeros ⊂ℨ on the CL ℒ⊂, which assumes the 3.9. The Approximation Formulas∀ ∊ ℝ: of the =[ Extremes] + {}; of[ζ] ∊ℕ,0≤and of the{ kth} <1 Order Moments on the CL RH to be true, the infinite sum is written: ∑ =−∑ Complex number in the CS By efficientlyIn ℂ, =+ =ℜ using the relation() + ℑ(ζ)(;∈ℂ;,∊=½+; ))/R(ζ( ))ℝ=(; tan)its= (conjugateω(/2))(, it̂) is = easier̅ =− ()( to)̅ numeri- =()() ⇒ |()| =1⇒() = cally estimate the curves ofIn the ℂ,ζ maximâ =1−=1−− ;,∊ on the CL,We and finally to derive obtain,ℝ; results taking from into them account in the the pairs of zeros on the CL (>0 and <0): Complex number 1− associated with ̅ ( ⁄) (⁄) CS. In addition, the ζ maxima,its conjugate being rare ̌ =1− events, = it iŝ =1−+ advisable′( to) . carry Γ out̂ 2 a π/4rotationΓ 2 1 1 1 () = = ; ()() =1 of the ζ function, in order̂ to make the three surfaces (real,½, imaginary 0 = lnΓ√(2 and⁄)−½ modulus)(1+½Γ(2⁄) of) + − + symmetric of with respect to (()). ̅ − + Critical strip (CS) the function In be the at theCS same: =ℜ level,() and+ ℑ to() make= + ; 0≤≤1,>0; ≠1 the estimation of extremes more. robust. 1 By multiplying by (=) if( ) +2 is not a zero: () = ()()() =()|()| The regular sampling of the ζ function from et /8, and not a regular sampling from , Critical line (CL) On the CL ℒ∶==½: =½+; >0. − ( ) \ ℒ∶=0≤≤1;≠½; ≠1 () ⁄ ℑ CS, except the CL |()| ≠0 ⇒This formula() = converges = slowly ⇒ on ( the) = ± imaginary|()| component, ⇒ because=tan of the terms in |()| ( ) 2 () = ∑ . In the CS, the function is divergent. ℜ Zeta function: . (⅝) One considers in this CS, the analytical continuation (). When is very large, we can do the approximation:( ) ( ≈)| ( ⅝)| ≈ . If is zero, Hospital’s rule gives: =− Êta Dirichlet function: () =()(1−2 ). The sum can then be broken down into two sums if ⋘ : Derivative of the function () =−∑ log()⁄ ′() ℜ () |()| =0 ⇒ () =− = ⇒ () = ±()⁄ ⇒− =tan ()( ) ∑ ( ) |′()| 1 1 ℑ (1) 2 = −log = + k-order derivatives of the function − − − In the CS, one uses the analytical continuationThe ofnon-trivial these derivatives zeros are at the breaking point of the two functions +⁄ and () =()(̂) =(−We) (can⁄1− . then) choose to obtain a good approximation, but the convergence is al- Functional equation of the function Γ(̂⁄)2 () =2 sin(⁄) 2 waysΓ(1− slow.) Using= ½ the power series expansion of Γ(2⁄), we serially expand () to find : Γ(2⁄) If was always very small compared to , which is never the case since |()| =1⇒(̅ ) = =() ; = ⁄ 4 −2{̃}; = −1⁄(24) ; () =()(̂) =sweeps()()̅ the= entire()( range); of ordinates, we could have an estimate of the infinite sum: ( ⁄) Functional equation on the CL Γ 2 () =3i 9 139i 1475 1 ln i − 1 1 73i 131 =1− Γ−(2⁄) + + +⋯1+ − − + +⋯ ⋘ ⇒ ≈ t=⁄(28) = 128⁄ 2 3072 32768 4 3 18 1620 9720 ̃ =ln(⁄)=⁄(2) ln(⁄)(2) . 1 1 + 2 + 2 ln 2 + () −24ln The anamorphosis produces a stronger Anamorphosis: ̃ i 1 (6×7×24−5= )i 7×24 −5 19 × 31 × 8707i 317 × 531103 elongation = of =1−the axis −as increases.− 4 − 6 − − 24 2! (24) 5×3!(24) 5×4!(24) 7×5!(24) 35 × 6! (24) We define by the adjective “digressive” a 13series × 5243712823i that exhibits1 1 a beginning31 × 661 × simi- 28660561 27 × 8197117457520809i lar to a known series, but which varies −noticeably with the≈ subsequent−1 + 2 + terms. 2 ln 2 + ′′(2−)−ln 5×7!(24) 24 5×8!(24) 25 × 11 × 9! (24) For example, the cosine series is written:81 ×cos 5099 = × 1 4517050132181 − + − +⋯+ − ! ! ! +⋯ Digressive series (−1) +ℴ(). We will then note, in order11×5×10! to simplify(24 the) writing, a ()! We can therefore extract the rotation ==−1⁄(24) from this expansion: digressive cosine: () =1− + − +⋯+(−1) + ! ! ! ()! = + () ℴ(). = + ()≈=−1⁄(24) Complex zeros ℨ in the CS = + ; () =0 =½+ ; () =0 is a three-order polynomial in = −⁄(4! ): Zeros ⊂ℨ on the CL ℒ⊂ The zeros are ordered in pairs, and 1−. Calculations are24 carried out with17807 >0 29 × 10079 43 × 11009087 13 131 × 23027 = 7 − 7 + − + − Lambert function The main branch is defined by:5 () = ⇔=7×2! ;≥−17×5×3!⁄ . 7×5×4! 7×5! 3458158927391587 7 × 47 × 2095343 × 148000093 Γ() = Gamma function + − +⋯ 3×5×7×9×11×6! 3×5×7×9×11×7! Euler’s constant Euler–Mascheroni constant: = lim− ln + ∑ ≈ 0.5772156649 → ()(168) Taylor series () = ∑ (− ) = 1 +! 5 Bernoulli numbers Coefficients of the power series4 of = ∑ ℬ36 + 31 × 12 + !(34 × 239 + 2 × 1643717 + 53 × 66797) 35 5 1 + (68 × 61 × 107923 × 965611 + 23 × 4881469 × 16227313) +⋯ 4511 We can deduce an approximate value of () on the CL:

168 ≈ (1+); = −1⁄(24) 5

168 168 () = ≈ + (1+)≈() 1 + ≈ ≈() 5 5

Information 2021, 12, x FOR PEER REVIEW 7 of 35

Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

2.5.5. Hadamard’s Product Formula Hadamard’s product formula contains in its expression, on the one hand, the non- trivial zeros and on the other hand, the functional Riemann equation. It therefore has a remarkable strength to characterize the zeta function and its deep relationship with neighborhoods of zeros. We know the formula: ′() 1 Γ′(1+½) 1 1 = ln2−½−1− −½ + + () −1 Γ(1+½) − (½) Moreover, we know that: ∑ =1+ ⁄ 2−ln2−½ln; = (1+½) (½) Information 2021, 12, x FOR PEER REVIEW 3 of 35 The formula becomes: ′() 1 1 = ln√ −½(1+½) − + () −1 − Table 1. Notations. Information 2021, 12, 22 Using a CL point of ℒ and the zeros ⊂ℨ on the CL25 ℒ⊂ of 35 , which assumes the Definition Information 2021, 12, x FOR PEER REVIEW Notation 18 of 35 RH to be true, the infinite sum is written: ∑ =−∑ Absolute value ∀ ∊ ℝ: || = > 0; − < 0 We finally obtain, taking into account the pairs of zeros on the CL (>0 and <0): ∀ ∈ ℂ: = + = = || (); also allows us to calculate with better( accuracy) the kth-order moments of the ζ function. ′ || =alternation|+|1= that generates+ ; ,1 , a ,non-trivial ∈ ℝ 1 zero in a restricted interval. On the CL, the calcu- We find the following formulas from the sample= ln√ data:−½ (1+½) + − + Complex number () =lations are( ⁄)̅extremely 2 simplified:− >+ 0; = (⁄) 1 + < 0; ∀ ∈ × 6 ∈ [ ] (=½+; =) (=)1+2+ () =()(̂) = ()()̅ =()() ⇒ |()| =1⇒() = n 1, 2 10 ; k 1, 8 : ζ =0: =; 2 i ⁄n, 2w, k > 0;− = −⁄ 2 < 0 1 = 2 + i2π(n − 2 + k/8In) /ℝW, the((n integer− 2 + k part/8)/ ande) the fractional part of a real number : Integer and fractional partsThis formula converges slowly on the imaginary component,Γ(̂⁄)2 becauseΓ(2⁄) of the terms in ∀ ∊ ℝ: =[] + {}; [] ∊ℕ,0≤{} <1 () = = ; ()() =1 1 . iπ/4 1 iπ/4 1Γ(2⁄) Γ(2⁄) MComplex0( ) = max number ζ in+ theiu CS = max R Ine ℂ, =+ =ℜζ + iu =(max) + ℑ (e);∈ℂ;,∊ζ + iuℝ; its(⅝ conjugate) ̅ =− u0; ≠1+ . ℜ − − − ℒ∶==½ =½+; >0 Critical line (CL) On the CL Table: 1. Notations. . ( ) =−( )|( )| We1 Rcan then 1 choose to obtain a goodIf approximation, is zero, Hospital’s but rulethe convergencegives: is al- Order 1CS, moment: except Mthe1( CL) = ζ + iu du ≈ −0.00661\ ℒ∶=0≤≤1;≠½; ≠1ln + 0.694 ln ln + 0.357 0 2 ways slow. Definition Notation 2() = ∑ . ′() ℜ () ( ) = 1 R 1 + |()|≈=0 ⇒ +( In) =− −the −CS, the ≈ function=− ⇒ is divergent.() = ± ()⁄ ⇒− =tan Order 2Zeta moment: function:M2 If was0Absolute ζalways2 iu valueverydu smallln compared2γ to1 ln , 2whichπ ln is∀ never ∊ ℝ: | the| = case since > 0; − < 0 |′()| () ℑ () 2 One considers in this CS, the analytical continuation . () 1.6834 sweeps the entire range of ordinates, we could have an estimate∀ ∈of ℂ: the =infinite + sum: = = || ; Êta Dirichlet function: 3 () =()(1−2 ). ⁄ 1 R 1 0.86375 The3 non-trivial zeros are at the breaking point of the two functions + and Order 3 moment: M3( ) = ζ + iu du ≈ 2 (ln ) + 16.53 || = |+| = + ; , , , ∈ ℝ 0 2 1 2π1 ⁄( ln) =− − 1∑ log()⁄ Derivative of the function⋘ Complex⇒ number ≈ − . 4 = (⁄) 2 > 0; 1 R 1 4 2() 4 3 Order 4 moment: M4( ) = ζ + iu du ≈ 1/ 2π (Using(ln) =) ∑the+ 0.123power−log(ln (series)) −expansion ( ⁄) of Γ(2⁄), we serially expand () to find : k-order derivatives of the function0 2 () = + < 0; 2 1 1 + 2 + 2 ln 2 + −24ln ( ) + − In the= CS, one uses the analytical continuation=0: =() of these⁄ 2 derivatives > 0; = −⁄ 2 < 0 0.748 ln 1.402 ln 0.816. |()| =1⇒() = = ; = ⁄ 4 −2{̃}; = −1⁄(24) ; 4 () =(In)6 (ℝ̂,) the= integer()(1− part) and the fractional part of a real number : Integer and fractional parts Γ(̂⁄)2 Functional equation of the function 1 1 3i 9 139i ∀475 ∊ ℝ:½ =[] + {}i; [] ∊ℕ,0≤1 {73i} <1 131 () =2 sin( ()⁄) 2 Γ(1−) = Complex number≈ 1 =1−in + the 2 +CS 2 ln− 2 + ′′In +2ℂ, −ln=+ =ℜ + ()+⋯1++ ℑΓ(2(⁄));∈ℂ;,∊− ℝ−; its conjugate + ̅ =− +⋯ 24 8 128 3072 32768 3 18 1620 9720 () =()(̂) =()()̅ In= ℂ(, )̂=1−=1−− ;,∊(̅ ); ℝ; Complex number 1− associated with Γ(2⁄) Functional equation on the CL ( its conjugate) ̌ =1− =̂̅ =1−+ . i 1() =6×7×24−5 i 7×24 −5 19 × 31 × 8707i 317 × 531103 = =1− − − ( ⁄) − − − 24 2! (24) 5×3!Γ 2(24̂ symmetric) 5×4! of ( with24) respect 7×5! to (½,(24 0). ) 35 × 6! (24) t=⁄(2) = ⁄ 2 Critical strip (CS) 13 × 5243712823i In the CS 31: ×=ℜ 661 ×( 28660561) + ℑ() = + ; 0≤≤1,>0; ≠127 × 8197117457520809i . ̃ =ln(⁄)=⁄−(2) ln(⁄)(2) . The− anamorphosis produces− a stronger Anamorphosis: ̃ Critical line (CL) 5×7!(24) On5×8! the CL(24 ℒ∶==½) : =½+; >025 × 11 × 9! (24.) elongation81 × 5099 of ×the 4517050132181 axis as increases. CS, except the CL − \ ℒ∶=0≤≤1;≠½; ≠1+⋯ We define by the adjective11×5×10! “digressive”(24 a series) that exhibits a beginning simi- () = ∑ . In the CS, the function is divergent. Zetalar function: to a known series, but which varies noticeably with the subsequent terms. ⁄( ) ( ) We can Onetherefore considers extract in this the CS, rotation the analytical ==−1 continuation24 from . this expansion: For example, the cosine series is written: cos = 1 − + − +⋯+ Êta Dirichlet function: = + () ()!=!()(!1−2 ). Digressive series Derivative(−1 of )the +ℴfunction( ). We will then note, in order to simplify() =− ∑the writing,log()⁄ a ()! = + ()≈=−1⁄(24) Figure 19. () Adjusted curve (in red) of the maximum of the modulus of the zeta function and data ( ) ∑ ( ) digressive cosine: () =1− + − +⋯+ =(−1)−log + (in cyan) of the moduli of thek-order 8 × 1.2 derivatives× 106 values of of the the zeta function moduli. ! ! ! ()! is a three-orderIn the CS, polynomialone uses the in analytical = −⁄ continuation(4! ): of these derivatives ℴ(). () =()(̂) =()(1−) Table5 shows the estimations for higher even24 moments from 617807 to 12. 29 × 10079 43 × 11009087 13 131 × 23027 Complex zeros ℨ in the CS = 7 − 7 + = + −; () =0 + − ( ⁄) Functional equation of the function ½ Γ ̂ 2 5 =½+ 7×2! ; (7×5×3!) )=2=0 sin(⁄)7×5×4! 2 Γ(1−) = 7×5! k ( ⁄) R 2 (k+2)/4 C(k) 3458158927391587 7 × 47 × 2095343 × 148000093Γ 2 Table 5. M = ζ 1 + iu du = A(k) ln + B(k) for k = 3 to 6. Zeros ⊂2k ℨ on the0 CL 2ℒ⊂ The zeros+ are ordered in pairs,( ) =and−( 1−)(̂). =()()̅ =()(̅ ); +⋯ 3×5×7×9×11×6! 3×5×7×9×11×7! Calculations are carried out with >0 ( ⁄) Functional equation on the CL Γ 2 k 2k (k + 2)/4 M2k C(k) A(k) B(k) () = Lambert function The main branch is defined by: () = ⇔= ;≥−1Γ(⁄2 ⁄). 168 3 6 5/4 M6= 1 + 4 ( 0.67) −1040 t=⁄(2) = ⁄ 2 Gamma function Γ = 4 8 3/2 M8 4.625 3.34 −26,600 4̃ =ln(⁄)=36 ⁄(2) ln(⁄)(2) . The anamorphosis produces a stronger 5Euler’s constant 10 7/4Anamorphosis:Euler–MascheroniM10 ̃ 5.42 constant: = 11 lim −− ln640,300 + ∑ ≈ 0.5772156649 + 31 × 12→+ (elongation34 × 239 of + 2the × 1643717 axis as + increases. 53 × 66797 ) 6 12 2 M12 6.4035 26 ()(−5)15,184,200 Taylor series 1() = ∑ (−) We define by !the adjective “digressive” a series that exhibits a beginning simi- + (68 × 61 × 107923 × 965611 + 23 × 4881469 × 16227313 ) +⋯ 45lar11 to a known series, but which varies noticeably with the subsequent terms. FigureBernoulli 20 shows numbers the fitting of the first powerCoefficients moments. ofM the( power) and theseries numerical of = ∑ ℬ 0 ! WeFor can example, deduce thean approximatecosine series valueis written: of ( cos) on =the 1 −CL: + − +⋯+ values of the coefficients C(k) for the higher moments are significant for the Lindelöf ! ! ! 2k Digressive series 1 R 1 168ε conjecture which is equivalent to the statement: ζ +(iu−1 ) du =+ℴ( ∀)k. We∈ will then note, in order to simplify the writing, a 0 2 ()! ≈ (1+); = −1⁄(24) 5 ∀ > N, ε 0. digressive cosine: () =1− + − +⋯+(−1) + ! ! ! ()! 168 ( 168) () = ≈ + (1+)≈() 1ℴ + . ≈ ≈() 5 5 Complex zeros ℨ in the CS = + ; () =0 =½+ ; () =0 Zeros ⊂ℨ on the CL ℒ⊂ The zeros are ordered in pairs, and 1−. Calculations are carried out with >0 Lambert function The main branch is defined by: () = ⇔=;≥−1⁄ . ( ) Gamma function Γ = Euler’s constant Euler–Mascheroni constant: = lim− ln + ∑ ≈ 0.5772156649 → ()() Taylor series () = ∑ (−) ! Bernoulli numbers Coefficients of the power series of = ∑ ℬ !

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Information 2021, 12, x FOR PEER REVIEW 7 of 35

Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

1 1 ≈ 1 + 2 + 2 ln 2 + ′′(2)−ln 24

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Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

We can therefore extract the rotation ==−1⁄(24) from this expansion: = + ()

= + ()≈=−1⁄(24)

1 Figure 21. (a) A segment In of et starting at 8 showing the break of the term arg(ζ(s))/π at the zero location; (b ) is 50 a plots three-order polynomial in = −⁄(4! ): of et : we observe the properties of 2 adjacent equivalence classes. 24 17807 29 × 10079 43 × 11009087 13 131 × 23027 = 7 − 7 + − + − The similarity between both general formulas therefore appears:5 7×2! 7×5×3! 7×5×4! 7×5! 3458158927391587 7 × 47 × 2095343 × 148000093 1 1 + − +⋯ R∗( ) = t ln t/e − + 1 + 1; N∗( ) = t ln(t/e) − + 1 + arg(ζ(s))3×5×7×9×11×6!/π 3×5×7×9×11×7! a 8 8

168 1 = 1 + = − − = 1 + Knowing 2 ω 2 (α θ) π/8 π{t ln(t/e)} 1/(96πt) which5 is only dependent of = 2πt and knowing in which quadrant is located (a, b), we can4 infer the last 36 + 31 × 12 + (34 × 239 + 2 × 1643717 + 53 × 66797) term arg(ζ(s))/π: 35 5 1 + (168 × 61 × 107923 × 965611 + 23 × 4881469 × 16227313 ) +⋯ 1st, 4th quadrant : ω/2π; 2nd quadrant : ω/2π + 1 ; 3rd quadrant : −ω45/2π11− . 2 We can deduce an approximate value of () on the CL: 4.2. Heteroscedasticity and the Evolution of the Hierarchical Model According to n 168 ≈ (1+); = −1⁄(24) The RVs Ten are uniformly distributed along the axis et, of mathematical expectation 5 µen = n, and of slowly increasing standard deviation: 168 168 () = ≈ + (1+)≈() 1 + ≈ ≈() eσn = 0.12555 W(ln n) + 0.08856 5 5

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Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

Information2.5.5.2021 Hadamard’s, 12, 22 Product Formula 28 of 35 Hadamard’s product formula contains in its expression, on the one hand, the non- trivial zeros and on the other hand, the functional Riemann equation. It therefore has a 7 remarkable strength to characterizewith for small the zetan: ∀ nfunction< N = and10 its: 0.23 deep< relationshipeσn < 0.346. with neighborhoods of zeros. We know theThe formula: difference between the minimum and maximum probability densities is of the ′() order of: 1 Γ′(1+½) 1 1 = ln2−½−1− −½et(pd fmax) − et(pd+ fmin) < 2(0.033+ ln(n ) + 0.7 ± 0.2). () −1 Γ(1+½) − 7 ∀ < ∈ [ − + ] The Gaussian RV Ten is experimentally truncated(½) at keσ: n 10 : Yen n keσ, n keσ ; Moreover, we know that: ∑ =1+ ⁄ 2−ln2−½ln;∼ 20 = (1+½) k ≈ 4; 8eσ <2.77. For n < N, N = 10 ,( the½ numerical) value of the standard deviation The formula becomes:σ on the transformed axis et allows the overlapping of two consecutive RVs on each in- e h i 1 1 ′() terval eIn = n − 2 , n + 2 . Thus,1 with the extension1 to 8eσ in Ce of the RV Ten, the RVs Ten ( ) = ln√ −½ 1+½ − + th () and Ten+1 overlap and some−1 realizations − of two consecutive RVs are switched. The n I I I Using a CL point zeroof ℒ canand lie the in zeros one of⊂ℨ the three on the intervals CL ℒ⊂en−,1 ,whichen, en+ assumes1. Therefore, the it is possible that zeros can be ahead (in the interval eIn−1, in 6.5% of cases), that zeros are late (in the interval RH to be true, the infinite sum is written: ∑ =−∑ eIn+1 in 6.5% of case), but in general (87%of cases) the zeros are “on time” in the interval ∼ >0 <0 We finally obtain, takingeIn = into[µn −accokσunt, µn the+ k pairsσ]; k =of 1.5.zeros on the CL ( and ): ( ) With such standard deviations for the RV, it often happens, including for small n ′ 1 1 1 = ln√(for −½ example(1+½n = 136) +), that− both the (n − 1)th+ and nth zeros or the nth and (n + 1)th zeros () ̅ − + are in the same intervaleIn. The1 model then facilitates the calculation of the number of zeros less than= ( a value) +2with a possible error of −1, 0, +1. In this distribution model, the RVs − Ten are considered as independent. It is thus possible (and this happens theoretically in This formula converges slowly on the imaginary component, because of the terms in 1.84% of cases for n < N) that the RV Ten+1 is located before the RV Ten. Thus, the (n + 1)th . zero attached to RV Ten+1 arrives ahead and the nth zero attached to the RV Ten arrives late, (⅝) When is very large,so much we can so do that the a approximation: permutation in the ≈ arrival⅝ orders,≈ to which. the zeros in question are linked, is possible. With this assumption of independence of RVs Ten, the (n + 1)th zero The sum can then be broken down into two sums if ⋘: attached to the RV Ten+1 is in fact the nth according to the order of arrival on the axis. However,1 this assumption 1 of independence 1 does not identify where these permutations = + take − place. Knowledge − should be extracted − from an additional phenomenon, resulting from the morphogenesis of the ζ function, in order to pinpoint the permutations. In the We can then choose to obtain a good approximation, but the convergence is al- absence of additional knowledge of the mechanism which produces zeros, we can avoid ways slow. the obstacle of permutations in the arrival order, and be satisfied by considering a simpler If was always very small compared to , which is never the case since interpretation: the RVs are Gaussian and all pairwise independent, except two successive sweeps the entire range of ordinates, we could have an estimate of the infinite sum: Gaussian RVs Ten and Ten+1, which are dependent with the constraint Ten < Ten+1. With this parameterization,1 1 theln order − 1 n of arrival is always compatible with the number of the RV Yn. 100 ⋘ ⇒ When≈ nis very large n> N; N ≈ 10 , the Gaussian RV spreads more widely, so that 4 the1 mixing1 + of 2 RVs + 2 becomes ln 2 + more() −24ln and more frequent. The nth zero might lie within the = [ − + ] interval4 of n 2, n 2 6zeros, and we would need to consider more than two consecutive RVs to be greater or smaller than the value of the previous RV. Any permutations are then 1 1 more challenging to interpret. ≈ 1 + 2 + 2 ln 2 + ′′(2)−ln 24 4.3. The Limits of the Model It is crucial to determine the limits of a theoretical model. The current model was inferred from values of y < 1020. We cannot claim to generalize this model beyond this value without making some hypotheses or assumptions which must be justified. If the model seems robust in its architecture, we can discuss the legitimacy of the use of the full normal distribution, the independence of adjacent RVs and the heteroscedasticity of the sequence of normal laws, which proceeds from a more fragile induction. Calculations towards N = 10100 on a segment ∆n = 1010 are essential to finally being able to confirm these properties.

4.3.1. Questioning the Legitimacy of Statistics: Observation versus Genesis The second layer of the model implements RVs. However, the origin of zeros is concealed under experimental observation through a statistical ﬁlter. Needless to say, the successive zeros of the CL do not obey the morphogenetic assumptions of a random distribution, since the ζ function is deterministic. The occurrence of a zero is generated by the nullity coincidence of both members of the inﬁnite sum of trigonometric functions Information 2021, 12, 22 29 of 35

√ e−i(y ln n modulo 2π)/ n, and this simultaneity is not due to chance. The Laplace–Gauss law is one of the foundations of statistics, based on the law of large numbers summarizing the work of Laplace [25], Cebyšëvˇ [26] and Paul Lévy [27]. Statistics which focus more on the analysis of the RV’s average behavior, rather than on rare events, should be used with caution when considering the extremes of mathematical functions. RVs are assumed to be independent, but the local layout is no accident (Figure 22). The Poisson model [28], Information 2021, 12, x FOR PEER REVIEW 7 of 35 which is the basis for the analysis of series of independent observations, is therefore not a suitable model for the appearance of zeros, because of the undeniable local dependence of the various sub-series of adjacent zeros. If we accept the systematic dependence of two successive RVs Yn < Yn+1, the process can even be considered as Markovian since: P(Yn+1|Yn,..., Y1) = P(Yn+1|Yn), with the constraint yn+1 > yn.

Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

1 1 ≈ 1 + 2 + 2 ln 2 + ′′(2)−ln 24

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morphogenesis of the ζ function and perhaps even the effects on the order or arrangement of prime numbers. However, it must be remembered that the continuous values of the ζ function do not come from a random draw, and that therefore, if the nth zero is disturbed in its positioning, it is clear that the neighboring zeros are also concerned in a way that should be explored. Information 2021, 12, x FOR PEER REVIEW 3 of 35

4.3.4. Questioning Rare Events and the Emergence of Rogue Waves In order to analyze the extreme behaviors of the ζ function in the CS, we can use the statistical modeling of rare events, while knowingTable that 1. the Notations. statistical approach is delicate (Figure 23). Only variables derived from the ζ function which can be assimilated to the observation ofDefinition small variations in a natural phenomenon can be the subjectNotation of such an | | examination.Absolute An analysis value of the growth of the standard deviation∀ ∊ ℝ: of the= RV Ten >along 0; − n, < 0 based on the law of large numbers, allows us to obtain a law∀ of∈ ℂ: evolution. = + For = the study= || (); of maxima, it is more reasonable to use the results of the extreme|| = |+ value| = theory + of Fisher ; , , , ∈ ℝ and TippettComplex [29], which number states that, under certain conditions, = the distribution(⁄) 2 of the ex- > 0; tremes (minima and maxima) of a large number of independent = and identically(⁄) distributed + < 0; (i.i.d.) RVs converge to one of the following three distributions=0: = (Gumbel⁄ 2 [30], > Fr 0;échet = [ −31],⁄ 2 < 0 Weibull [32]): the parametric generalized extremeIn ℝ value, the integer (GEV) distributionpart and the family fractional is used part of a real number : Integer and fractional parts here. The analysis of the rare values on the minima and∀ maxima ∊ ℝ: = of[ the] + local{}; [ standard] ∊ℕ,0≤{} <1 deviationsComplex of thenumber variations in the inCS the appearance In ℂ, =+ =ℜ of zeros, estimated() + ℑ on( 10,000);∈ℂ;,∊ successiveℝ; its conjugate ̅ =− samples, gives some lessons, without the underlying assumptionsIn ℂ, of̂ =1−=1−− ;,∊ the statistical theories ℝ; Complex number 1− associated with being likely to be made in default. its conjugate ̌ =1− =̂̅ =1−+ . ̂ symmetric of with respect to (½, 0). Critical strip (CS) In the CS : =ℜ() + ℑ() = + ; 0≤≤1,>0; ≠1. Critical line (CL) On the CL ℒ∶==½: =½+; >0. CS, except the CL \ ℒ∶=0≤≤1;≠½; ≠1 () = ∑ . In the CS, the function is divergent. Zeta function: One considers in this CS, the analytical continuation (). Êta Dirichlet function: () =()(1−2). Derivative of the function () =−∑ log()⁄ ()() = ∑ −log() k-order derivatives of the function In the CS, one uses the analytical continuation of these derivatives Figure 23. Rogue wave, with very ﬂat waves around: n = 3,206,674 and n + 1 = 3,206,675.() =()(̂) =()(1−) Functional equation of the function Γ(̂⁄)2 () =2 sin(⁄) 2 Γ(1−) =½ Γ(2⁄) 4.4. Model Adherences () =()(̂) =()()̅ =()(̅ ); 4.4.1.Functional The Resemblance equation withon theCebyšëv’sˇ CL First Function Γ(2⁄) () = ˇ Cebyšëv’s ﬁrst function is given by: θ(x) = ∑p≤x ln p; p prime. Γ(2⁄) We know several bounds for this function, for example: t=⁄(2) = ⁄ 2 ̃ =ln(⁄)=⁄(2) ln(⁄)(2) . The anamorphosis produces a stronger Anamorphosis: ̃ ln ln k − 2.050735 elongation11 of the axis as increases. θpk ≥ k ln k + ln ln k − 1 + f or k ≥ 10 ; pk : kth prime number lnWek define by the adjective “digressive” a series that exhibits a beginning simi- The three-term asymptotic expansionlar to of a the known Lambert series, function but which is as kvariestends noticeably to +∞: with the subsequent terms. For example, the cosine series is written: cos = 1 − + − +⋯+ 2! ! ! ! Digressive series ln lnk ln ln k W(k) = ln k − ln ln k(+−1) ++ℴ(). We will then note, in order to simplify the writing, a ln(k )! ln k digressive cosine: () =1− + − +⋯+(−1) + ! ! ! ()! The connection between the heteroscedasticity of normal distributionsℴ( and the). evolution of the distribution of prime numbers is immediate: θpk ≥ kW(k) + 2 ln ln k. Complex zeros ℨ in the CS m = + ; () =0 We deduce for the primorial pm# de pm : pm# = ∏k=1 pk; ln pm# = θpm ≥ mW(m). =½+ ; () =0 Zeros ⊂ℨ on the CL ℒ⊂ The zeros are ordered in pairs, and 1−. Calculations are carried out with >0 Lambert function The main branch is defined by: () = ⇔=;≥−1⁄ . ( ) Gamma function Γ = Euler’s constant Euler–Mascheroni constant: = lim− ln + ∑ ≈ 0.5772156649 → ()() Taylor series () = ∑ (−) ! Bernoulli numbers Coefficients of the power series of = ∑ ℬ !

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4.4.2. The Rapprochement with the Erdös–Kac Theorem Probabilistic number theory explicitly uses probabilities to state results on the distribution of properties of natural numbers. ω(n) being the number of prime factors distinct from a non-zero natural number n, we know the Hardy–Ramanujan theorem [33]: “The normal order of the number ω(n) of prime factors distinct from a number n is ln(ln n)”. We also know the more precise Erdös–Kac theorem [34]: √ n o R λ − 2 ∀λ ∈ R : lim 1 n ≤ x|ω(n) ≤ ln ln x + λ ln ln x = √1 e t /2dt . x→∞ x 2π −∞ This theorem expresses that the number of prime factors distinct from a natural number n for 1 < n < N tends to a normal distribution with mean ln ln N and variance ln ln N as N tends to inﬁnity. When running the Eratosthenes sieve to identify prime numbers, which are checked only once, ω(n) is the number of scores of each natural number. Information 2021, 12, x FOR PEER REVIEW This number is maximum for primorials (e.g., 7# = 2 ×73 of× 355 × 7 = 210 ⇒ ω(210) = 4).

Thus, we can generate the numbers n < N = 107 with ω(n) prime numbers which follow a more or less bell-shaped histogram: 3 on average, very often (2, 4), less frequently (1, 5), more rarely (6, 7) and only one occurrence for 8: ∀n ≤ N = 107 : 1 ≤ ω(n) ≤ 8. For example:

n = 2 × 3 × 5 = 30; n = 22 × 133 × 292 < N ; ω(n) = 3.

m n power of a prime number, n = pk ⇒ ω(n) = 1:

2 2 p437 = 3049; 3049 < N; ω(p437) = ω p437 = 1

n n = p = m p ⇒ (n) = m primorial, m# ∏k=1 k ω : Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more p8# = 2 × 3 × 5 × 7 × 11 × 13 × 17 × 19 < N; ω(p8#) = 8 than the Logarithm function. To generate all the natural numbers n < N, there is an economy principle with 2.5.5. Hadamard’s Product Formulathe parsimonious use of ω(n) prime numbers {bn} drawn from the elements of the base 2 Hadamard’s product formulaBk{p1, p contains2,..., pk} in; p kits 0 Applications and <0): Finally, the model brings theoretical and practical advantages and disadvantages for ′() 1 1 1 = ln√computational−½(1+½ applications:) + − + () ̅ − + • This offers an easy way1 to calculate the numerical value for the seed of the nth zero: = () +2 the pivot w5,n is indeed very close to the result. − • The passage through the Fourier transform is clearly improved since the zeta function This formula converges slowly on the imaginary component, because of the terms in in the Ce space is quasi-periodic, with period 1. It is a fairly regular corrugated sheet . and the local Fourier coefﬁcients are then crystallized on a few frequencies, thanks to (⅝) When is very large, we can do the approximation: ≈ ⅝ ≈ . the resemblance and the correspondence between the Gaussian pdf of a local zero and the local morphology of the zeta function. The sum can then be broken down into two sums if ⋘: 1 1 1 = + − − − We can then choose to obtain a good approximation, but the convergence is always slow. If was always very small compared to , which is never the case since sweeps the entire range of ordinates, we could have an estimate of the infinite sum: 1 1 ln − 1 ⋘ ⇒ ≈ 4 1 1 + 2 + 2 ln 2 + () −24ln = 4 6

1 1 ≈ 1 + 2 + 2 ln 2 + ′′(2)−ln 24

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Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

2.5.5. Hadamard’s Product Formula Hadamard’s product formula contains in its expression, on the one hand, the non- trivial zeros and on the other hand, the functional Riemann equation. It therefore has a remarkable strength to characterize the zeta function and its deep relationship with neighborhoods of zeros. We know the formula: ′() 1 Γ′(1+½) 1 1 Information 2021, 12,= ln2−½−1− 22 −½ + + 32 of 35 () −1 Γ(1+½) − (½) Moreover, we know that: ∑ =1+ ⁄ 2−ln2−½ln; = (1+½) (½) The formula becomes: • The model has the drawback of being statistical. ′() • It does not provide1 answers on questions1 about the ordering and the numbering of = ln√ −½(1+½) − + ()Information 2021these, 12, x zeros:FOR PEER it isREVIEW necessary−1 to deepen− the morphological engine of the zeta function 3 of 35 and the genesis of the folds of this function. ℒ ⊂ℨ ℒ⊂ Using a CL point of •and theThe zeros model is valid on overthe CL alarge interval, whichn assumes< 1020 ,the but must be validated for much RH to be true, the infinite sum is written: ∑ =−∑ higher values. We finally obtain, taking into account the pairs of zeros on the CL (>0Table and 1. <0Notations.): 5. Conclusions ′() Definition1 1 1 Notation ( ) = ln√ −½ TheAbsolute1+½Information article+ value 2021 analyzes−, 12, x FOR numerically PEER REVIEW+ the global and local∀ ∊ distributionℝ: || = of > the 0; − non-trivial < 0 18 of 35 () ̅ − + ∞ −s zeros of Riemann’s zeta function ζ(s) = ζ(x + iy) = a + ib = ∑ n along the CL() n o 1 ∀ ∈ ℂ: = +n= 1 = = || ; ( ) 1 =L X =+2; > 0 of the complex plane. The behavior of the ζ function in the CS 2 −C || = |+| = + ; , , , ∈ ℝ (in fact,Complex its analytical number extension, since the ζ function diverges = in( the⁄) CS) 2 is governed >in 0; alternation that generates a non-trivial zero in a restricted interval. On the CL, the calcu- This formula converges slowlyparticular on the by imaginary the fractional component, part t becauseof the of variable the termst = inϕ (y) (= ⁄)y/(2π) ln(y/(2πe)) = e lations are extremelye = simplified: + < 0; . ( ) = + = =0: =⁄ 2 > 0; = −⁄ 2 < 0 t ln t/e et et ; t y/2π. ( ) ⅝ When is very large, we can do the approximation:=½+; ≈L ⅝()≈=(). (̂) =()()̅ =()() ⇒ |()| =1⇒() = The n non-trivial zeros n on obey In theℝ, partitionthe integer of part the variable and the etfractionalas follows: part of a real number : Integer and fractional parts [ ] { } [ ] { } The sum can then be broken down into two sums if ⋘ : ∀ ∊ ℝ: = + ; ∊ℕ,0≤ <1 1 Γ(̂⁄)2 Γ(2⁄) Complex number∀n ≥ in1 :the CS= + i ; Inζ (ℂ, )=+ =ℜ= 0 ⇒ R(ζ( ())) + ℑ= (ζ)(;∈ℂ;,∊)) = 0 ℝ; its conjugate ̅ =− 1 1 n 2 n 1 n () = n = n ; ()() =1 In Γℂ(2, ⁄)̂ =1−=1−− ;,∊Γ(2⁄) ℝ; = + Complex − number 1− −associated with − t = ϕ( ); t + ε = n − 2 ; ε = −1, 0,its+ conjugate1; α = π /4̌ =1−− 2πt =̂̅ =1−+ . en n en By multiplying byn () if is noten a zero: () = ()()() =()|()| We can then choose to obtain a good approximation, but the convergence ̂ issymmetric al- of with respect to (½, 0). iθn 3 ways slow. θn = −1/(4! n); ωn = αn + arg e + i(1 +(iθ=ℜn))θn4!7/5() + ℑ+ ···() = + ; 0≤≤1,>0; ≠1≈ αn + θn ℑ() Critical strip (CS) |()| ≠0 ⇒ In the() CS= : = ⇒ () = ±|()|⁄ ⇒ =tan. If was always very small compared to , which is never the case since|()| ℒ∶==½ =½+; >0ℜ() 2 CriticalI line (CL) R 0 On the CL : 0 2 . sweeps the entire range of ordinates, we (coulζ( )d) have an( estimateζ ( n)) of the infiniteωn sum:α n + θn ζ ( n) limCS, except the= CL− = tan( ) ≈ tan \ ℒ∶=0≤≤1;≠½; ≠1; ξ( n) = − 0 → R(ζ( )) ( 0( )) If2 is zero, Hospital’s2 rule gives:| ζ( ()|) =−( )| ( )| n ζ n n 1 1 ln − 1 () = ∑ . In the CS, the function is divergent. Zeta function: ⋘ ⇒ ≈ This article shows that the zeros are distributed on the CL according to a two-layered ( ) 4 One considers′() in this CS, the( ) analytical( )continuation⁄ ℜ (). M |()| =0 ⇒() =− = ⇒ = ± ⇒− =tan hierarchical model . () |′( )| ( ) ( )( ) 2 1 Êta1 Dirichlet + 2 + 2 function: ln 2 + −24ln = 1−2 th. ℑ () = The ﬁrst, deterministic layer deﬁnes two pivots w1,n and w5,n, linked to the n zero, 4Derivative of the 6function () =−∑ log()⁄ ⁄ close to the cancellation of the imaginary andThe realnon-trivial parts of zeros the ζ function,are at the and breaking involves point the of the two functions + and () ⁄ (1)+=W∑(t/e) −log() 1 1 k-orderLambert derivatives function, of the via anfunction anamorphosis − ϕ: ϕ. (y) = et ⇔ y = 2πe e. ≈ 1 + 2 + 2 ln 2 + ′′(2)−ln InUsing the CS, the one power uses seriesthe analytical expansion continuation of Γ(2⁄) ,of we these serially derivatives expand () to find : 24 () =1 ()1(̂) =()(1−) ∀n ≥ 1 : w n = 2πt n/W(t n/e); t n = n − 2 +(); ζ + i w n 1, 1, |()| 1,=1⇒1,() = =8 ;2 = ⁄1, 4 −2{̃}; =( −1⁄)⁄(24) ; Functional equation of the function ½ Γ ̂ 2 = εn(1 + i(48 (+)···=2)) sin(⁄) 2 Γ(1−) = w1,n Γ(2⁄) 3i 9 139i 475 i 1 73i 131 =1− − + (+) =()(+⋯1+̂) =()()̅ =− ()(̅−); + +⋯ ∀n ≥ 1 : = 2πt8/W128(t /e);3072t = n −327682 + 5 ; ζ 1 + i 3 18 1620 9720 Functional equation on wthe5,n CL 5,n 5,n 5,n 8 2 wΓ5,(n2⁄) () = = en((−48 w5,n + ···) + i) Γ(2⁄) i 1 (6×7×24−5)i 7×24 −5 19 × 31 × 8707i 317 × 531103 = =1− − − t=⁄−(2) = ⁄ 2 − − |ε |, |e24| < ε;2! ∀(24n <)10 7 ⇒5×3!ε < 1/(24n ) 5×4!(24) 7×5!(24) 35 × 6! (24) n n ̃ =ln(⁄)=⁄(2) ln(⁄)(2) . The anamorphosis produces a stronger Anamorphosis: ̃ 13 × 5243712823i 31 × 661 × 28660561 27 × 8197117457520809i − elongation− ofw the axis as− increases. W is the main branch of the Lambert5×7! function:(24W)(z) = w ⇔5×8!we (=24z ; )z≥ −1/e . 25 × 11 × 9! (24) th The pivot w5,n turns out toWe be81 a define × good 5099 estimateby × the 4517050132181 adjective of the ordinate “digressive” of the an serieszero: that exhibits a beginning simi- − +⋯ lar to11×5×10! a known series,(24 but) which varies noticeably with the subsequent terms. 5 ∗ 2π n − 2 + ∗ a ∀n ≥ 1 : = = For8 ;example,− the< cosine ;series(a < 10is written: for n < Ncos= 10 =7) 1; − + − +⋯+ n w5,n 5 We n can ntherefore extract the rotation ==−1! !⁄(24! ) from this expansion: n−2+ ln n Digressive series W 8 e(−1 ) = +ℴ+ (() ). We will then note, in order to simplify the writing, a ()! digressive cosine: ()==1− ++ −()≈=−1+⋯+(−1⁄()24 ) + = + | !| ≈ 2!πn ≈!2πn ()! An asymptotic estimate for the zero of rank n, zn xn iyn is n W(n/e) ln n . ( ) The second, stochastic layer precisely is aallocates three-order the polynomial zeros and involves ℴin = a −. sequence⁄(4! ): of Complex zeros− 1ℨ in the CS = + ; () =0 RVs Yn = ϕ Ten . To express24 the features17807 of the RVs 29Yn ×∼ 10079(µn, σn), it is43 wiser × 11009087 to work 13131 × 23027 = 7 − 7 + − =½+ +; () =0 − = ( ) Zerosin the ⊂ anamorphicℨ on the CL plane ℒ⊂Ce 5 ϕ C . In this space7×2!CeThe, after zeros7×5×3! a change are ordered of variable in pairs,7×5×4! from yandto e t,1−. 7×5! the distribution of zeros along the transformed CL follows the realization of a stochastic 3458158927391587 7 × 47 × 2095343>0 × 148000093 + Calculations are− carried out2 with +⋯ process of independent Gaussian RVs, regularly spaced: Ten ∼ N µn = n, σn ; σn = Lambert function The3×5×7×9×11×6! main branch is defined by:e 3×5×7×9×11×7!() = ⇔=e e ;≥−1⁄ . Γ() = Gamma function 168 = 1 + Euler–Mascheroni constant: = lim− ln + ∑ ≈ 0.5772156649 Euler’s constant 5 → 4 36 ()() Taylor series + 31 × 12 + (34( ×) 239= ∑ + 2 × 1643717(−) + 53 × 66797) 35 5 ! 1 = ∑ ℬ Bernoulli numbers + (68Coefficients × 61 × 107923 of the × power 965611 series+ 23of × 4881469 × 16227313 ) +⋯ 4511 ! We can deduce an approximate value of () on the CL: 168 ≈ (1+); = −1⁄(24) 5

168 168 () = ≈ + (1+)≈() 1 + ≈ ≈() 5 5

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Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more than the Logarithm function.

We can therefore extract the rotation ==−1⁄(24) from this expansion: We use here the natural = order+ of zeros,() that is to say that we admit, in this formula, the dependence between two consecutive RVs ( + > ). n 1 n • The common law of the maximum of the absolute= value of+ ζ( )(in) its≈=−1 real and imagi-⁄(24) nary components, and in absolute value is a three-order polynomial in = −⁄(4! ): 24 1 17807 29iπ/4 × 100791 43 × 11009087iπ/4 1 13 131 × 23027 M0 = = max7 ζ − 7+ iu + = max R−e ζ + iu + = max I e ζ +−iu 5 u

168 168 () = ≈ + (1+)≈() 1 + ≈ ≈() 5 5

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Table 1. Notations.

Definition Notation Absolute value ∀ ∊ ℝ: || = > 0; − < 0 ∀ ∈ ℂ: = + = = || (); || = |+| = + ; , , , ∈ ℝ

Complex number = (⁄) 2 > 0; Figure 1. Comparison between the Lambert and Logarithm functions: Lambert’s function compresses high values more = (⁄) + < 0; than the Logarithm function. =0: =⁄ 2 > 0; = −⁄ 2 < 0 In ℝ, the integer part and the fractional part of a real number : 2.5.5. Hadamard’s Product FormulaInteger and fractional parts ∀ ∊ ℝ: =[] + {}; [] ∊ℕ,0≤{} <1 Hadamard’s product formula Complexcontains in number its expression, in the CS on the one In hand,ℂ, =+ =ℜ the non- () + ℑ();∈ℂ;,∊ℝ; its conjugate ̅ =− trivial zeros and on the other hand, the functional Riemann equation. It therefore has a In ℂ, ̂ =1−=1−− ;,∊ℝ; Complex number 1− associated with remarkable strength to characterize the zeta function and its deep relationship with neigh-its conjugate ̌ =1− =̂̅ =1−+ . borhoods of zeros. We know the formula: ̂ symmetric of with respect to (½, 0). ′() 1CriticalΓ′ strip(1+½ (CS)) 1 In1 the CS : =ℜ() + ℑ() = + ; 0≤≤1,>0; ≠1. = ln2−½−1− −½ + + () −1CriticalΓ line(1+½ (CL)) − On the CL ℒ∶==½: =½+; >0. CS, except the CL (½) \ ℒ∶=0≤≤1;≠½; ≠1 Moreover, we know that: ∑ =1+ ⁄ 2−ln2−½ln; = (1+½) (½) () = ∑ . In the CS, the function is divergent. Zeta function: The formula becomes: One considers in this CS, the analytical continuation (). ( ) ′ Êta Dirichlet function:1 1 () =()(1−2 ). = ln√ −½ (1+½) − + () Derivative of the −1 function − () =−∑ log()⁄ () Using a CL point of ℒ and the zeros ⊂ℨ on the CL ℒ⊂, which assumes the () = ∑ −log() k-order derivatives of the function RH to be true, the infinite sum is written: ∑ =−∑ In the CS, one uses the analytical continuation of these derivatives () =()(̂) =()(1−) We finally obtain, taking into account the pairs of zeros on the CL (>0 and <0): Information 2021, 12, 22 Functional equation of the function 34 of 35 Γ(̂⁄)2 () =2 sin(⁄) 2 Γ(1−) =½ ′() 1 1 1 Γ(2⁄) = ln√ −½(1+½) + − + () ̅ − + () =()(̂) =()()̅ =()(̅ ); 1 k Γ(2⁄) = Functional() +2 equation1 R on 1the CL • The moments Mk( ) = ζ + iu du: () = 0 − 2 Γ(2⁄) t=⁄(2) = ⁄ 2 This formula converges slowly on the imaginary component, because of the terms0.86375 in 3 M1 ≈ −0.00661 ln + 0.694 ln ln + 0.357;̃ =lnM3 (⁄)≈ =⁄(2ln) ln(+⁄)(16.532) . The anamorphosis produces a stronger . Anamorphosis: ̃ 2π2 (⅝) elongation of the axis as increases. When is very large, we can do the approximation: ≈ ⅝ ≈ . M ( ) ≈ ln + 2γ − 1 −ln 2π = ln − 1.68344 2 We define by the adjective “digressive” a series that exhibits a beginning simi- The sum can then be broken down into two sums if ⋘ : 1 4 3 lar to a known2 series, but which varies noticeably with the subsequent terms. M4( ) ≈ (ln ) + 0.123 (ln ) − 0.748(ln ) + 1.402 ln − 0.816 1 2π2 1 1 For example, the cosine series is written: cos = 1 − + − +⋯+ = + ! ! ! 2k + − Digressive −R series − k 2 k e(k) • M ( ) = 1 + iu du =(−14) ( )+ℴ(( ). We will then note, in order to simplify the writing, a The moments 2k 0 ζ 2 ln()! ln : We can then choose to obtain a good approximation, but the convergence is al- digressive cosine: () =1− + − +⋯+(−1) + ways slow. e(3) = 1.072; e(4) = 0.624; e(5) = 0.418; e(6) = 0.396 ! ! ! ()! If was always very small compared to , which is never the case since ℴ( ). sweeps the entire rangeThe of ordinates, modelComplex also we makes coul zerosd ithave possibleℨ an in theestimate to CS spot of a the family infinite of local sum: profiles of theζ = function + ; () =0 on the CL, around the RVs, according to the realization of the neighboring Gaussian =½+ RV. ; ( ) =0 1 1 ln − 1 ⋘ ⇒This family≈ canZeros be extended ⊂ℨ to on the the entire CL ℒ⊂ CS, thanks to the homothetyThe of thezeros morphologies are ordered in pairs, and 1−. 4 of theζ function in the CS. If the value n deviates a little from the meanCalculationsµn of the are RV carried out with >0 1 1 + 2 + 2 ln 2 + () −24ln Yn, the ζ function profile is regular around this neighborhood: the real and imaginary() = ⇔=;≥−1⁄ = Lambert function The main branch is defined by: . waves are4 almost stationary with6 sinusoids of similar amplitudes. On the other hand, if the ( ) Gamma function Γ = value n is significantly different from the mean of the RV Yn, a “rogue wave” emerges, in 1 1 ∑ ≈the vicinity1 + 2 of + 2the ln value2 +Euler’s ′′(2=)−ln µconstantn, worthy of the RoaringEuler–Mascheroni Forties, which contributes constant: = to the lim− ln + ≈ 0.5772156649 24 → development of local maxima of ζ(s) on the CL and in the CS (for x = 0). This very high()() Taylor series () = ∑ (−) wave is then accompanied by waves of tiny amplitudes, in dead calm, which cause the ! emergence of close consecutiveBernoulli numbers zeros. Coefficients of the power series of = ∑ ℬ ! Finally, the hierarchical model allows us to explain the absence of multiple roots,

by dissecting Hadamard’s product formula, with two close zeros. The numerical study of the Gaussian RV Ten. must be continued, in particular the slow evolution of its heteroscedasticity, which must be validated for high values of n; n > N ≈ 10100. Indeed, the very slow growth of the standard deviation would be a major property of the second layer of the model, which would reﬂect the local disorder of non-trivial zeros, −10 notably in their distributions for very low probability densities (< 10 ) of Ten, analyzable by the theory of extremes, and in relation to the distribution of prime numbers for very high values. The ζ function contains secrets that we can unravel through new analysis grids. The digital way offers to openly illustrate drafts of research. In this article, the Lambert function, by its elongation, reveals the uniform description of folds of the Zeta function along the transformed imaginary axis and the normal distribution locally distributes the zeros around regularly spaced pivots. Lambert’s unfolding standardizes the global aspect of the Zeta function in equivalent paradigms of length 1, but the stress relief of Gauss’s law, along n, expresses a progressive degree of freedom of the local position of the zeros, thanks to the increasing standard deviation in ln(ln n), most certainly, in relation with the Erdös–Kac theorem and with the maximum elasticity (perceptible on twin prime numbers) of the distribution of prime numbers for very high values of n.

Funding: This research received no external funding. Informed Consent Statement: Not Applicable. Data Availability Statement: The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request. The data are not publicly available due to the fact that these data are the result of calculations carried out by a wide range of specific application programs, in Python language. Acknowledgments: C Fagard-Jenkin, student of mathematics at St Andrews University, acted as an initial proofreader before submission. Conflicts of Interest: The author declares no conflict of interest. Information 2021, 12, 22 35 of 35

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