Numerical Calculations to Grasp a Mathematical Issue Such As the Riemann Hypothesis

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Article Numerical Calculations to Grasp a Mathematical Issue Such as the Riemann Hypothesis

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

Received: 3 April 2020; Accepted: 23 April 2020; Published: 26 April 2020

Abstract: This article presents the use of data processing to apprehend mathematical questions such as the Riemann Hypothesis (RH) by numerical calculation. Calculations are performed alongside graphs of the argument of the complex numbers ζ(x + iy) = a + ib and ξ(x + iy) = p + iq, in the 1 critical strip. On the one hand, the two-dimensional surface angle tan− (b/a) of the Riemann Zeta y y function ζ is related to the semi-angle of the fractional part of 2π ln 2πe and, on the other hand, the Ksi function ξ of the Riemann functional equation is analyzed with respect to the coordinates (x, 1 x; y). − The computation of the power series expansion of the ξ function with its symmetry analysis highlights the RH by the underlying ratio of Gamma functions inside the ξ formula. The ξ power series beside the angle of both surfaces of the ζ function enables to exhibit a Bézout identity au + bv c between ≡ the components (a, b) of the ζ function, which illustrates the RH. The geometric transformations in complex space of the Zeta and Ksi functions, illustrated graphically, as well as series expansions, calculated by computer, make it possible to elucidate this mathematical problem numerically. A ﬁnal theoretical outlook gives deeper insights on the functional equation’s mechanisms, by adopting a computer–scientiﬁc perspective.

Keywords: Riemann Hypothesis; numerical calculation; functional equation; congruence

1. Introduction This article presents the use of data processing to apprehend mathematical questions such as the Riemann Hypothesis (RH) by numerical calculation. The geometric transformations in complex space of transcendental functions, illustrated graphically, as well as series expansions, calculated by computer, make it possible to elucidate this mathematical problem numerically. The article sets out its reasoning using descriptive graphics. The mathematical calculations have been developed and checked by computer, and the graphics have been performed through extensive speciﬁc Python programs. The RH [1] (“non-trivial zeros are located exclusively on the critical line x = 1/2”) is illustrated here with mathematical notions and numerical calculations: power series expansions of the Gamma function and Ksi function of the Riemann’s functional equation, and calculations on complex numbers and angles between two surfaces or two curves. The whole study is focused in the critical strip . S The study contribution follows from the examination of the two-angle conjunction of the Zeta function and the Ksi function. On the one hand, the Zeta function has a remarkable property on the critical line: The real/imaginary ratio of the Zeta function is interpreted as the tangent of an angle. On the other hand, the Ksi function is congruent, within a homothety and a rotation, and to a remainder series, whose sum function angle is not symmetrical with respect to the critical line. On the contrary, the residual congruent series intrinsically contains this symmetry on the critical line. It is therefore essentially an angular functional constraint of the holomorphic Zeta function and of the associated Ksi function, which is at the origin of the property of the RH. This constraint enigma is contained in the Gamma functions’ ratio of the Ksi function.

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Information 2020, 11, 237 2 of 31 Ksi function, which is at the origin of the property of the RH. This constraint enigma is contained in the Gamma functions’ ratio of the Ksi function. The nine-pagenine-page article,article, in in German, German, by by B. B. Riemann Riemann [1 ],[1], is essentialis essential in thein the history history of mathematics of mathematics and isand instructive is instructive in understanding in understanding the conjecture’s the conjecture’s context. context. Historical Historical articles articles by Hadamard by Hadamard [2], Vall [2],ée PoussinVallée Poussin [3], Bohr [3], and Bohr Landau and Landau [4], Hardy [4], [ 5Hardy], and Hardy[5], and and Hardy Littlewood and Littlewood [6] allow us[6] toallow grasp us the to extentgrasp ofthe the extent question’s of the question’s diﬃculty. difficulty. Deligne’s demonstrationDeligne’s demonstration in 1973 [7 ,in8], 1973 via algebraic[7,8], via geometry,algebraic geometry, gives the visiongives the and vision the victorious and the pathvictorious in the path case ofin ﬁnitethe case sets. of A finite thoughtful sets. A introduction thoughtful introduction to the properties to the of theproperties Zeta function of the Zeta is the function book by is EC the Titchmarsh: book by EC 1951 Titchmarsh: edition [ 91951], 1986 edition edition [9], [ 101986]. Less edition fundamental [10]. Less booksfundamental retrace books the various retrace works the various up to theworks years up 1974to the [11 years] and 1974 1985 [11] [12 and]. A 1985 deep [12]. insight A deep of analytic insight numberof analytic theory number concepts theory and concepts methods and can method be seens incan the be 2004 seen book in the by 2004 Iwaniec book and by Kowalski Iwaniec [and13]. TheKowalski search [13]. for non-trivial The search zeros for non-trivial using the properties zeros usin ofg complexthe properties analysis of orcomplex number analysis theory is or presented number intheory historical is presented and recent in historical publications and [recent14–20]. publications [14–20]. After this introduction, Section2 2 presents presents thethe methodology,methodology, i.e.,i.e., thethe graphicgraphic andand computationalcomputational illustration strategy. Section Section 33 presentspresents preparatorypreparatory calculationscalculations thatthat areare essentialessential forfor thethe results.results. Section4 4 presents presents thethe resultsresults ofof thethe calculationscalculations byby graphicalgraphical representationsrepresentations andand powerpower expansions.expansions. Section 55 presents presents thethe discussiondiscussion onon proofproof strategies,strategies, beforebefore thethe conclusion,conclusion, wherewhere thethe resultsresults areare summarized and research roadmapsroadmaps areare presentedpresented moremore abstractly.abstractly.

2. Methods This articlearticle presentspresents aa relation relationϕ of of the the function functionζ(s )=()ζ=(x +(iy+) = a) +=+ib in a congruencein a congruence form au + bv c, with a and b being the real and imaginary parts of ζ(s), respectively,( ) and, u and v being form ≡ + ≡ , with and being the real and imaginary parts of , respectively, and, independentand being functionsindependent of Zeta, functions which of are Ze dependentta, which are only dependent on x and yonly. This on congruence and . This has itscongruence source in the Riemann’s functional equation ζ(s) = ξ(s)ζ(1 s)( and) the Gamma( ) function Γ whose variations has its source in the Riemann’s functional equation− =() 1− and the Gamma function Γ andwhose dependencies variations and are investigated.dependencies The are study investigated is based. onThe the study speciﬁcities is based of on the the Zeta specificities function, the of Ksithe functionZeta function, and the the Gamma Ksi function function, and by the analyzing Gamma them function, with respectby analyzing to the aforementionedthem with respect relations. to the Theaforementioned components arelations.and b, formed The ofcomponents ripples, cancel and each other, formed out periodically of ripples, and cancel alternately, each other and thusout causeperiodically on the criticaland alternately, line, in the and neighborhood thus cause ofon these the undulations,critical line, thein the emergence neighborhood of an additional of these zeroundulations, but, this the time, emergence common of to an both additional curves. Thezero article but, this thus time, provides common a numerical to both curves. illustration The article of the Riemannthus provides Hypothesis a numerical (RH). illustration of the Riemann Hypothesis (RH). The methodology ﬂowflow chart is summarized inin FigureFigure1 1::

Figure 1. Methodology Flow chart of numerical calculationscalculations to grasp the RH issue.

2.1. Strategy for Discovering Fertile Domains and Sterile Domains to Zeros We try to identify, in a domain s = x + iy, s C , a function ϕ of the complex plane which We try to identify, in a domainD ≔{=+,∈ℂ∈ }, a function of the complex plane ( ) = 2 obeys:which obeys:s ∀ ∈ ⊂ℂ∶C : ϕ ζ∘s ()0=0. Expressed. Expressed in inR ℝ, the, the requirement requirement is is as as follows: follows: wewe trytry toto ∀ ∈ D ⊂ n ◦ 2 o 2 identify, in a domain M = (x, y) R a pair (u, v) of real functions in R which obeys: M identify, in a domainD ≔{=(∈, ) ∈ℝ } a pair (, ) of real functions in ℝ which∀ ∈obeys: D ⊂ R2 : (ζ(s)) u(x, y) + (ζ(s)) v(x, y) = 0. ∀ ∈< ⊂ ℝ ×∶ ℜ()×= (, )×+ ℑ()×(, ) =0. This domain is a fertile domain where it is possible to ﬁnd non-trivial zeros, since (ζ(s)) = 0 This domain D is a fertile domain where it is possible to find non-trivial zeros, since< ℜ()= then entails (ζ(s)) = 0, and vice-versa. This function pair (u, v) must obey the following requirements: 0 then entails= ℑ()=0, and vice-versa. This function pair (, ) must obey the following Therequirements: pair (u, v) mustThe notpair result (, ) from must the Zetanot function,result from nor fromthe anZeta auxiliary function, expression nor from of Zeta, an auxiliary but must onlyexpression depend of on Zeta, the but point must coordinates only depend of the on domain; the point this coordinates pair must alsoof the not domain; cancel outthis simultaneously pair must also

Information 2020, 11, 237 3 of 31 for any given point, or more precisely, can only cancel out simultaneously for a zero measure set of points in this domain. To solve the RH, it is necessary to show both that the favorable domain is the critical line and also that there are no such pairs in the other domains, in particular in the critical strip S, L apart from the line , (zone ). We thus propose to find pairs of functions in the critical strip, L S\L such as au + bv = c by favoring the pairs (u, v) that cancel out c. It is clear that the Riemann functional equation offers tracks for mapping out the research roadmap, but it is not sufficient. Indeed, the functional equation expresses a powerful necessary constraint: if the point M s or (x, y) is a non n o trivial zero, then Mˆ sˆ = 1 s or (1 x, y) , M s = x iy or (x, y) and Mˆ sˆ = 1 x + iy or (1 x, y) − − − − − − − are also zeros. The pair (u, v) that we are looking for must therefore meet this requirement and exhibit this symmetry. It is hence advisable to decompose the critical strip into longitudinal sections of double lines x = xp and x = 1 xp, given the functional equation symmetry. Conceived with this constraint, − the problem then becomes finding pairs up, vp for all points P with coordinates sp = xp + iy from the lines x = xp such that: y : ζ sp up(y) + ζ sp vp(y) = 0. ∀ < × = × y : ζ sˆp uˆp(y) + ζ sˆp vˆp(y) = 0. ∀ < × = × uˆp, vˆp is the symmetric pair, associated with the pair up, vp . If this search for lines perpendicular to the real axis proves unfruitful, it is advisable to entrench oneself towards a search for purely isolated points xp, yp in the critical strip which satisfies this identity. On the other hand, to denounce the RH, it suffices to find at least one point Q such that: Q xq, yq : ζ sq uq + ζ sq vq = 0; uq , 0; vq , 0 ∃ < × = × 2.2. Qualification of Domains by Filtering Modulo a Reference Series This strategy can only succeed by the determination of an intrinsic property of the Zeta function, which can generate an equivalence relation in order to fold the imaginary axis on itself in a quotient space, so as to avoid calculations to infinity. It is through the search for angles and congruent series that this simplification is achieved. In our case, we decompose the Ksi function into irreducible series and we calculate the congruent function associated modulo this reference series. This method finally has the advantage of taking into account the Zeta function constraints through the Ksi function. The process allows us to move away from the Zeta function, by emphasizing the Ksi function instead, which is easier to analyze, as a two Gamma function ratio. The whole process takes place thanks to Information Technology. In the critical strip, the theoretical approach therefore consists in decomposing the Ksi function of the functional equation into amplitude and phase elements. The phase is broken down into a power series. This power series is filtered modulo a power series in (x 1/2) so as to dissolve the axis − of symmetry.

2.3. Illustration Strategy The illustration strategy of the article here consists in modifying the Zeta function by successive transformations that are linear combinations (function of s) of the real part a and the imaginary part b so as to reach a zero-function target au + bv = 0. Theoretical approaches use universal and powerful mathematical instruments and sometimes neglect the intrinsic numerical properties of the Zeta function, the Ksi function of the Riemann functional equation or the Gamma function. Indeed, certain elements of these functions are not easily elucidated by mathematical tools, such as for example the fractional part of variables or the angle oriented in complex space. Numerical computation, whilst incapable of demonstrating anything formally, opens theoretical horizons and bridges an important gap in mathematical analysis, because it Information 2020, 11, 237 4 of 31 Information 2020, 11, x FOR PEER REVIEW 4 of 32

2.4.readily Extensive crowbars Use of itself Power into Series even Expansions the most stubborn of functions, unveiling their mysteries. It is the subsidiarityThere are insu manyfficiency works of theon Zeta,numerical Ksi and methods Gamma for functions Zeta function that this zeros: article The attempts computer to rectify.made it possible to calculate specific values with high precision [21]. In this article, we place ourselves at the 2.4. Extensive Use of Power Series Expansions confluence of numerical approaches and theoretical formulations: We calculate and visualize curves and surfaces,There are which many we works illuminate on numerical with power methods series. for ZetaIndeed, function the polynomial zeros: The form computer of a function made it possibleremains tomore calculate intuitive specific than values its conventional with high precisionmathematical [21]. Inexpression. this article, For we example, place ourselves the Gamma at the function’sconfluence local of numerical behavior approachesis less easily and understood theoretical in formulations:the integral theoretical We calculate form and visualize curves than and surfaces,2 which⁄ ( we⁄)½ illuminate1 + with+ power− series.⁄ + Indeed, the polynomial form of a function in the form ( ) ( ) . Moreover, our approach is above all remains more intuitive than its conventional! mathematical! expression. For example, the Gamma a question of evaluating the influence and the symmetry of the abscissa under theR configuration ∞ tz 1e tdt function’s local behavior is less easily understood in the integral theoretical form 0 − − than of several variables ( =, ½ =−½, =1−) compared to its ordinate . The state of fusion, z 1/2 1 1 139/5 1 in the form √2π/e(z/e) − 1 + + + . Moreover, our approach is above all or on the contrary of separability,12 zof the2!( 12variablesz)2 − 3!(12 inz)3 a formulaO z4 is also more practical to apprehend a question of evaluating the influence and the symmetry of the abscissa x under the configuration of when this appears under the configuration of power series expansions. The originality of this article 1 stemsseveral from variables this reconciliationx = x, x1 2 between= x /2 ,numericalx = 1 xcalculationcompared and to itsthe ordinate formula yreadability.. The state of fusion, 0 / − 1 − or on the contrary of separability, of the variables in a formula is also more practical to apprehend when2.5. Mathematical this appears Notation under the configuration of power series expansions. The originality of this article stems from this reconciliation between numerical calculation and the formula readability. Where possible, classical mathematical notations have been selected. To lighten the writing, it is necessary2.5. Mathematical to define Notation some original notations.

2.5.1.Where Notations possible, for the classical Zeta Function mathematical notations have been selected. To lighten the writing, it is necessary to deﬁne some original notations. Simplification of writing a ratio: = 1⁄(2) = 12;=1 ⁄(24) = 1⁄ 24 2.5.1. Notations for the Zeta Function Integer part and fractional part of1 a real: =[] + {} ;1 ∈ ℝ; [] ∈ℕ;0≤{} <1 Simpliﬁcation of writing a ratio: 2π = 1/(2π) = 1/2π; 24y = 1/(24y) = 1/24y IntegerThe domains part and and fractional associated part points of a real: are apresented= [a] + a in; Tablea R; 1[ aand] NFigure; 0 a2. < 1 { } ∈ ∈ ≤ { } TheReal domains part, imaginary and associated part of points a complex: are presented =+= in Table ℜ(1 )and+ Figureℑ();∈ℂ;,∈ℝ2. Real part, imaginary part of a complex: s = x + iy = (s) + i (s); s C; x, y R Conjugate of a complex number: ̅ =−; ̅ ∈ℂ < = ∈ ∈ Conjugate of a complex number: s = x iy; s C − ∈ SymmetricSymmetric of of a a point point s: sˆ: =̂ 1=1−s=1−−s = 1 x iy − − − OppositeOpposite of of a a point point s (conjugates (conjugate of theof the symmetric symmetric point): point):sˇ = 1̌ =1−s=sˆ == 1̂̅ =1−+x + iy − −

FigureFigure 2. 2. CriticalCritical strip strip and and critical critical line line on on the the Cℂ plane, with the four associated points.points. Table 1. The four associated points: M, Mˆ , M, Mˇ . Table 1. The four associated points: , ,, . M := s = (x, y) M := s = (x,y) ∶= =(,−)− ∶==(,) Mˆ∶=:= 11 − s = =( (11 −x, ,y −)) ∶=M ˇ :∶== M(ˆ1−:=(1) =(1−,)s) = (1 x, y) − − − − − Homothety of the imaginary axis : =⁄ 2 HomothetyCritical strip: of the∶=0≤≤1;≠1 imaginary axis y: t = y/2π Critical strip: := 0 x 1; s , 1 A point in theS critical≤ strip:≤ = + ; ∈ ℂ, ≠ 1; , ∈ ℝ; 0 < < 1 ; > 0 A point in the critical strip: s = x + iy ; s C, s , 1; x, y R; 0 < x < 1 ; y > 0 Critical line: ℒ∶==½ ∈ ∈ Critical line: := x = 1/2 A point on theL critical line: = ½ + ; ∈ ℝ Critical strip, except the critical line: \ ℒ∶=0≤≤1;≠½; ≠1

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A point on the critical line: 𝓈 = 1/2 + iy ; y R ∈ Critical strip, except the critical line: := 0 x 1; x , 1/2; s , 1 P s S\ L ≤ ≤ Zeta function: ζ(s) = ∞ n . n=1 − Dirichlet eta function: η(s) = ζ(s) 1 2 s+1 . − − In the critical strip, the series ζ(s) is divergent; the analytical extension by η(s) is used. The function Ksi is written: ξ(s) = 2sπs 1 sin(πs/2)Γ(1 s). − − The Zeta function satisﬁes the functional equation: ζ(s) = ξ(s)ζ(1 s) = ξ(s)ζ(sˆ). − 2.5.2. Anamorphosis of the y Axis It produces a stronger elongation of the y axis as y grows: y y Anamorphosis of y: ey = log = t(log t 1) 2π 2πe − 2.5.3. Digressive Series u˘ 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. For example, the cosine series is written: 2 4 6 2n cos x = 1 x + x x + ... + ( 1)n x + ℴ x2n+1 . We will then note, in order to simplify the writing, − 2! 4! − 6! − (2n)! a digressive cosine:

2 4 6 2n a2x a4x a6x n a2nx + cos˘ (x) = 1 + + ... + ( 1) + x2n 1 − 2! 4! − 6! − (2n)!

We obviously require the sequence an to have “nice” properties so as not to degrade the meaning h i of proximity with the cosine function. The notation cos˘ only recalls the behavior analogous to a cosine, but it is clear that it is not the cosine function. In the same form, we will note, in a mnemonic way, the digressive exponential with its argument:

a x a x2 a x3 a xn e˘x = 1 + 1 + 2 + 4 + ... + n + (xn); arg(e˘x) = x˘ 1! 2! 3! n!

e˘iθ = cos˘ θ + i sin˘ θ

Pay attention throughout the remainder of this article to the modulus and argument (angle) in question. Indeed: ix ix iε2 ix If e˘ = e + ε1e ; then arg e˘ = x˘ = x + ε3; but ε3 , ε2

→ → If e˘ix = eix + ε eiε2 = 1 then ε 2 + 2ε eix.eiε2 = 0 and e˘ix = ei(x+δx) k k k 1 k 1 1 Again, these mnemonic notations are simply a notational convention and only refer to a speciﬁc power series and have no computational meaning.

2.5.4. Symmetric Polynomial, Symmetric Series

In this article, we call a symmetric polynomial P around x = 1/2 in x and (1 x), a polynomial − such that x : P(x) = P(1 x); for example, polynomials of the type: ∀ − X n n n P(x) = an x + (1 x) + bn(x(1 x)) − −

A sequence an , symmetrical around a C, obeys a similar property: h i ∈ X n X n z C : f (a + z) = ∞ an(a + z) = f (a z) = ∞ an(a z) ∀ ∈ n=0 − n=0 −

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3. Materials and Preparatory Calculations It is assumed that the reader is already familiar with the fundamentals of mathematics. There are many books that the reader can consult: a classic or standard book, which can be recommended to the reader interested in the specific subjects, is proposed in each following paragraph. It is crucial to understand the behavior of the Zeta, Ksi and Gamma functions in the critical strip, in particular the values for which the real and imaginary parts cross zero, and the angles between the two parts, because they play an essential role in the RH. In this article, we use basic theorems to dissipate the illustration’s complexity. We therefore use the following theorems, results and formulas: Bézout’s theorem, Stirling’s formula, Taylor’s formula, and operations on complex numbers. In order to facilitate the reading of the following sections, preparatory calculations by computer are grouped in this section. The reader can skip the development of the calculations at first reading, although they bring together the formation of the tools necessary for reasoning and understanding latter sections. In addition, this section represents a more substantial part of the computer work of this article. The study of the Ksi function’s numerical behavior requires power series expansions as a function of y, assuming that x is small compared to y. These computational developments require rigor and significant efforts; the calculations were checked by computer. We use the Python language [22] and the SciPy ecosystem [23] for all the computations and we visualize them using the matplotlib library [24]; we use mpmath [25] as an arbitrary-precision substitute for Python’s float/complex types and for functions (Gamma, Zeta), with arbitrary precision.

3.1. Bézout’s Identity The book [26] is intended as an introduction to the subject of Commutative Algebra, not a substitute for the treatise such as Bourbaki. A geometrical viewpoint of the Bézout’s theorem and convergent series rings is provided in the book of Chenciner [27]. Bézout’s identity is the result of elementary arithmetic, which proves the existence of solutions to the linear Diophantine equation: au + bv = gcd(a, b) of unknown integers u and v, where a and b are relative integer coeﬃcients and where gcd(a, b) is the greatest common divisor of a and b. Let a and b be two relative integers ( Z) and their gcd(a, b) = d; then there are two integers u and v such ∈ that au + bv = d. Étienne Bézout generalized this result, notably to polynomials, and more generally the result can be stated on any principal ring. This theorem echoes Hilbert’s zero theorem (Hilbert’s Nullstellensatz). Let K be an algebraically closed ﬁeld, let a , ... , am K[X , ... , Xn] be polynomials 0 ∈ 1 without common zeros. Then there exists u , ... , um K[X , ... , Xn] verifying a u + ... + amum = 1. 0 ∈ 1 0 0 Our approach, adjacent to this identity, consists in trying to write the functional equation ζ(s) = a + ib = ξ(s) ζ(sˆ) = (p + iq) aˆ + ibˆ = paˆ qbˆ + i pbˆ + qaˆ in the form of two equations − au + bv = c and i(au0 + bv0) = ic0. ζ and ξ are then series, and we study the divisibility of these series and their congruence with reference to a pivotal series.

3.2. Taylor Function Standard text books on Mathematical Analysis are those by Rudin [28] and especially by Whittaker and Watson [29], which contain a solid mathematical discussion of the transcendental functions. If the function f with complex values is diﬀerentiable in z up to the order n 1, then the ≥ Taylor–Young function is written as:

(2) (n) f 0(z) f (z) f (z) f (z + h) = f (z) + h + h2 + ... + hn + ℴ( hn) 1! 2! n!

The main part of the power series expansion from f to z to order n is the polynomial Pn:

X (k) n f (z) k Pn(z + h) = h k=0 k!

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The Taylor series from f to z is deﬁned as the power series whose nth partial sum is equal to Pn, for all integers n. The Taylor series of f at this point z is the series of functions:

X f (n)(z) f (z + h) = ∞ hn n=0 n! This series is used for theoretical reasoning. However, the nth-order expansions are numerically used to reach numerical accuracy. In this article, the functions are holomorphic and indefinitely differentiable. The power series expansions makes it easier to find the angles between real and imaginary surfaces. The following power series expansions are used:

a(a 1) a(a 1)..(a n + 1) (1 + z)a = 1 + az + − z2 + ... + − − zn + ℴ( zn) 2! n!

z2 zn ez = 1 + z + + ... + + (zn) 2! n! z2 z4 z2n cos z = 1 + ... + ( 1)n + z2n+1 − 2! 4! − (2n)! z3 z5 z2n+1 sin z = z + ... + ( 1)n + z2n+2 − 3! 5! − − (2n + 1)! During the calculations, compound series appear. These series are speciﬁcally the seeds of divisions of power series. Some are listed below:

2 3 4 5 6 7 8 z 7z 3z 11z 13z z 17z 19z e− ( 3 + 2z) = 3 5z + + + + ... − − − 2 − 2 24 − 120 48 − 5040 40320

2 3 4 5 6 7 8 z 2 15z 35z 21z 33z 143z 13z 17z e− 1 4z + 4z = 1 + 3z + + + ... − − − − 2 6 − 8 40 − 720 336 − 2688 3 4 5 6 7 8 9 1 z 3z 2 5z z 7z z z z 11z e− ( 2 + z) = 1 + z + + + ... − 2 − − 2 − 12 8 − 240 180 − 1120 8064 − 725760 3 4 5 6 7 1 z 2 2 10z 11z 49z 17z z e− 2 8z + 3z = 1 + 3z 5z + + + ... − 2 − − − 3 − 8 120 − 180 56 In this article, we use power series expansions extensively, because they are simple to interpret. On the other hand, they are long and complicated to calculate for the Gamma function and the Ksi function, especially when we try to highlight and understand their behavior according to the abscissa x under the conﬁguration of several variables x = x, x1 = x 1/2, x = 1 x with respect to its 0 /2 − 1 − ordinate y.

3.3. Power Series The reader can obtain more information about series and Complex Analysis in References [30,31], where a comprehensive treatise on holomorphic functions is given. P n A power series a (series of positive integer powers) is a series of functions of the form: anz h i where an is a complex sequence. Series is associated with its radius of convergence R( a ) = h i P n + h i sup z z C, anz converge > R + . The indefinitely derivable and expandable functions {| | ∈ } ∈ ∪ { ∞} P n in power series at point z0 are written in the neighborhood of point z0 as sum f (z) = n∞=0 anz of the power series of the variable z z , and this is then their Taylor series. A function of the complex − 0 variable, defined on an open set U (subset of a topological space which contains no point of its border), is said to be analytical on U when it admits a power series expansion in the neighborhood of every point of U. Such a function is indefinitely differentiable on U and called holomorphic. The product of two power series is defined, using the Cauchy product of series with complex terms:

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X X X Xn X ∞ n ∞ n ∞ n ∞ n anz bnz = akbn k z = cnz n=0 n=0 n=0 k=0 − n=0 R( c ) in f (R( a ), R( b )), h i ≥ h i h i P n P n P n In this case, the division of two power series is: n∞=0 anz = n∞=0 cnz / n∞=0 bnz P n P n P n The series are then congruent: ∞ anz ∞ cnz mod ∞ bnz n=0 ≡ n=0 n=0 The main parts, to the nth order, are also congruent: An(z) Cn(z) mod Bn(z) ≡ 3.4. Bernoulli’s Numbers and Bernoulli’s Polynomials

Bernoulli’s numbers n are involved in the Gamma function and in the Ksi function. These are B n t = P t the coeﬃcients of the power series of et 1 n∞=0 n n! . − B 1 1 1 1 = 1; = 1/2; i f k > 0 : + = 0 ; = ; = ; = ; = ; B0 B1 − B2k 1 B2 6 B4 − 30 B6 42 B8 − 30 B10 = 5 ; = 691 ; = 7 ; = 3617 ; = 43867 . 66 B12 − 2730 B14 6 B16 − 510 B18 798

n 1 n 1 2(2n)! 2n = ( 1) − 2n ; 2n ( 1) − B − |B | B ∼ − (2π)2n

n(x) are the Bernoulli polynomials: B X cos(2kπx nπ/2) n(x) = 2n! ∞ − B − k=1 (2kπ)n

2 3 2 (x) = 1; (x) = x 1/2; (x) = x x + 1/6; (x) = x 3x /2 + x/2; B0 B1 − B2 − B3 − (x) = x4 2x3 + x2 1/30; (x) = x5 5x4/2 + 5x3/3 x/6. B4 − − B5 − − 3.5. Group of Similarities: Homothety and Rotation of a Complex Function Let z be a complex number. In this article, we consider homotheties, with center O and ratio k , k R : (z) = kz. We also consider the rotations of a holomorphic function. These are rotations ∈ H R with center O and angle θ: (z) = zeiθ. Homotheties and rotations in the complex plane constitute R a subgroup of the group of similarities. y (x 1/2) − − In this article, we consider a homothety of ratio k = 2π We also use four rotations of center O and respective angles:

2 π 1 (x 1/2) θ1 = ; θ2 = 2π ey ; θ3 = ; θ4 = − . 4 − −24y − 2y

The rotations of the Zeta function have the property of preserving the position of non-trivial zeros: ζ(s) = 0 ζ(s) eiθ = 0. ζ(s) , 0 ζ(s) eiθ , 0. In the case of non-trivial zeros, L’Hôpital’s ⇒ × ⇒ × rule applies: 0 = ξ(s) 0 ζ (s) = ξ(s) ζ (1 s) . × ⇒ 0 × 0 − 3.6. The Gamma Function For further details on the Gamma function, the reader is referred to the historical book of Artin [32]. The Bernoulli–Euler–Legendre Gamma function is: Z ∞ z 1 t Γ(z) = t − e− dt 0

The Gamma function can be written in a power series form: z 1/2 z+µ(z) Stirling’s formula: z C Z∗ : Γ(z) = √2π z − e− ∀ ∈ \ − Information 2020, 11, 237 9 of 31

P 2k Binet’s function: µ(z) = ∞ B k=1 2k(2k 1)z2k 1 − − A property of the Gamma function is in particular: Γ(z) = Γ(z). The Gamma function quickly takes very low values, as y increases in the critical strip: Information 2020, 11, x FOR PEER REVIEW 68 9 of137 32 Γ(1/2 + i100) = ( 1.09179 + 1.04964i) 10 ; Γ(1/2 + i200) = (3.88183 8.28654i) 10 . − × − − × − In order to analyze the local behavior of the Gamma function, it is preferable to consider the Γ()⁄|Γ()| since it is the angle between both surfaces or curves that comes into play (Figure 3). The ( ) ( ) functionGamma Γfunctionz / Γ z intervenes since it is by the the angle values between 2⁄ bothand surfaceŝ⁄2 with or the curves values that for comes in intothe interval play (Figure [0,1]3).. The Gamma function intervenes by the values s/2 and sˆ/2 with the values for x in the interval [0, 1].

Figure 3. The Gamma standardized surface (real part and imaginary part), in the critical strip. Figure 3. The Gamma standardized surface (real part and imaginary part), in the critical strip. The The surfaces cancel each other out in turn and almost periodically. The functional equation uses the surfaces cancel each other out in turn and almost periodically. The functional equation uses the variable s/2 for Gamma. variable 2⁄ for Gamma. Undulations of the Gamma Function in the Critical Strip Undulations of the Gamma Function in the Critical Strip It is important to grasp the behavior of the Gamma function with x in the interval [0, 1] to identify the constraintsIt is important of the to Zeta grasp function. the behavior The Gamma of the function Gamma behaves function similarly with toin thethe complex interval function[0,1] to identify= ( )the= constraintsz = ( + of)( xthe+iy )Zeta function. The Gamma function behaves similarly to the complex f : f z z x iy and( plays) a major role in the RH. Indeed, in Stirling’s approximate function ∶=() = = (+z 1 ) and plays a major role in the RH. Indeed, in Stirling’s formula Γ(z) √2π/z (z/e) e 12z , the element √z plays an essential role, because it disrupts the ≈ Γ(z) ≈ 2⁄ (⁄) symmetryapproximate in theformula function ξ between s and sˆ = 1, thes. element √ plays an essential role, because it disrupts the symmetry in the function between− and ̂ =1−. In the critical strip x , 1, y , 0, the Gamma function is never canceled out. At constant y, ≠ 1, ≠ 0 the argumentIn the critical (angle) strip of Γ varies along, xthe. If Gamma the surfaces functionΓ(x, yis) arenever displayed, canceled the out. angle At constant of the Gamma, the Γ Γ(, ) functionargument ripples (angle) out of between varies alongπ and π. alongIf the thesurfacesy axis, but it is are important displayed, to the note angle that theof the ripples Gamma are −− obliquefunction to ripples the axis outx. between The Gamma functionand along is therefore the a axis, function but it that is important locally nests to note the variablesthat the ripplesx and yareinextricably. oblique to the axis . The Gamma function is therefore a function that locally nests the variables and inextricably. 3.7. The Ksi Function of the Functional Equation 3.7. The Ksi Function of the Functional Equation The Ksi function (Figure4) has remarkable properties that should be elucidated if we want to understandThe Ksi the function RH. The (Figure Ksi function 4) has remarkable manages the properties symmetry that in sshouldand sˆ =be1 elucidateds of the Zetaif we function: want to − ζunderstand(s) = ξ(s)ζ (thesˆ). TheRH. functional The Ksi relationship,function manages presented the insymmetry various forms,in isand as follows:̂ =1− of the Zeta function: () = ()(̂). The functional relationship, presented in various forms, is as follows: (s) (1 2 s ) ζ ( ) s Γ /( /2⁄) ξ(s) = = π Γ ½−− 2 () =ζ(sˆ) =Γ(1/2)Γ(s/2) (̂) Γ(½)Γ(2⁄)

Information 2020, 11, 237 10 of 31 Information 2020, 11, x FOR PEER REVIEW 10 of 32

Figure 4.4. Above:Above: KsiKsi functionfunction in in the the critical critical strip: strip: the the real real part part and and the imaginarythe imaginary part cancelpart cancel each othereach outother periodically out periodically and alternately. and alternately. Below: Below: Ksi modulus Ksi modulus and argument. and argument. The Ksi The modulus Ksi modulus is equal is to equal 1 for xto= 1 for1/2. =½ The surface. The surface obliquity obliquity (angle (anglearg(ξ) ) is small:()) is Thesmall: undulations The undulations are not are strictly not strictly parallel parallel to the realto the axis. real The axis. drawing The drawing might might make make the reader the re thinkader thatthink the that angle the angle does notdoes depend not depend on x: thison is: this not true.is not Thetrue.x ,Thearg( ξ,) linkage() linkageis very weak,is very but weak, is nevertheless but is nevertheless responsible responsible for the RH. for the RH.

However, toto analyze analyze the the synchronization synchronization of theof zeroingthe zeroing of the of real the and real imaginary and imaginary parts, the parts, relation the canrelation be written can be symmetrically:written symmetrically: Z Z ∞ ⁄ ∞ X 2 ⁄ F(s()=) =θ ()(t)ts/2 1dt== ∞ e πn tts/2 1dt − − − 0 0 1

s/2⁄ sˆ/2 ̂⁄ F(s())==ζ( s()Γ)(Γs(/22⁄))π− ==F(sˆ)(=̂) =ζ(sˆ)(Γ̂)(Γsˆ/(2̂⁄))2π− It isis essentialessential toto focus focus on on the the various various components components of of this this function functionξ to observeto observe the formationthe formation and gestationand gestation of this of symmetry.this symmetry. The The complex complex formation formation of symmetry of symmetry is in is fact in fact located located at the at corethe core of the of Gammathe Gamma function, function, which which intermingles intermingles the variables the variablesx and y andso vigorously so vigorously that it is that naive it is to naive hope to to hope split allto split the components all the components of this function of this function into both into independent both independent parts, one parts, function one offunctionx, and theof other, and of they. s 1/2 Γ(sˆ/2) otherThe of set-up. used in this article is: ξ(s) = π − . The function ξ, which is within a factor, Γ((̂⁄)s/2) a GammaThe set-up function used ratio, in this intermingles article is: the( variables) = ½x and. yThe, but function the local angle, which between is within both a factor, surfaces a (⁄) (realGamma and function imaginary) ratio, shows intermingles a lesser degreethe variables of dependence and compared, but the local to x angle. between both surfaces (real Onand the imaginary) critical line, shows the a ratiolesser becomes degree of the dependence ratio of two compared conjugated to Gamma. functions, and the modulus of the function ξ is 1. The function ξ then behaves like undulations which vary between 1 On the critical line, the ratio becomes the ratio of two conjugated Gamma functions, and the− andmodulus1. It isof not, the however,function a perfectis 1. The classic function trigonometric then behaves function. like The undula functionaltions which relation vary distorts between the relationship−1 and 1. It betweenis not, however, the Gamma a perfect functions, classic and trigonom therefore,etric thefunction. result The between functiζonal(s) and relationζ(sˆ), distorts outside the criticalrelationship line. between In the Ksi the formula, Gamma the functi ratioons, of and Gamma therefore, functions the result severely between destroys, () along and the(̂)x, axis,outside the the obliquity critical ofline. the In undulations the Ksi formula, of the the Gamma ratio of function, Gamma but functions this does severely not disappear destroys, completely. along the This axis, isa the primary obliquity result of of the this analysisundulations of the of Ksi the function. Gamma In function, addition, but the inversethis does of thisnot functiondisappear is: 1 ξcompletely.(s)− = ξ(sˆ This). In is the a primary decomposition result of of this Ksi intoanalysis elements, of the weKsi must function. be able In addition, to ﬁnd the the inverse inverse elements of this andfunction better is: understand () = ( hoŵ). In symmetry the decomposition works. of Ksi into elements, we must be able to find the inverse elements and better understand how symmetry works.

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3.7.1. Serial Expansion of Ksi One of the major elements of this article is the power series expansion of the Ksi function (if x is small compared to y).

s 1/2 √s µ(sˆ/2) µ(s/2) ξ(s) = (2πe) − P(s)Q(s); P(s) = s ; Q(s) = e − √ssˆ

Indeed, it is with the symmetry study in x and 1 x around x = 1/2 that the RH can be grasped. − The power series expansion is composed of several pieces: a main part P, that of an expression in s, multiplied by an exponential part Q of lower importance, but more sophisticated, with the intervention of the Binet formula of Bernoulli terms. The calculation of the power series expansion of this formula is very laborious, but this calculation makes it possible to exhibit the angle between the real and imaginary parts. This angle depends on x and y, and this connection occurs differently between s and sˆ. It is this dissimilarity that prevents finding a Bézout relationship in the critical strip, except on the critical line. The power series expansion consists of the product of two factors P and Q. s A first expression corresponds to the polynomial extracted from the expression √s/ √ssˆ .

√iy √1 ix/y P(s) = − s ys √1 (x(x 1)/y+2i(x 1/2))/y − − − s 1/2 1 /2 1 2 ix x(x 1) 2i(x /2) − = eiπ/4 y / s 1 1 − + − − − y − y2 y

x = x; x1 = x 1/2; x = x 1 0 /2 − 1 − p p !1/2 !! s/2 iy 1 ix0/y ix x x 2ix1/2 − P(s) = − = eiπ/4 y1/2 s 1 0 1 0 1 + q s − 2 s − y − y y y 1 x x /y + 2ix1 /y − 0 1 /2 The second expression Q corresponds to the exponential of Binet’s formula:

Q(s) = exp(∆µ)

X 2k 1 2 − 2k 1 2k 1 2k ∆µ = µ(sˆ/2) µ(s/2) = ∞ B sˆ − s − − k=1 2k(2k 1) − − We can extract the expression for the power series expansion when x is small compared to y:   X ( )k+1( )2k 1 !1 2k !1 2k 1 2/y − 2k  ix − i(1 x) −  ∆µ = i ∞ − B  1 + 1 + −  k=1 2k(2k 1)  − y y  − The expression ∆µ is symmetrical in x and x 1: The inverse of Ksi comes under the expression − ∆µ. The formula of Ksi then becomes: −

1/2 s iπ y − iq(x,y) ξ(s) = e 4 p(x, y)e 2πe

3.7.2. Expansion of the Main Part p

s !1/2 ! ix x(x 1) 2i(x 1/2) − 2 p(x, y) = 1 1 − − − y − y2 − y Information 2020, 11, 237 12 of 31

This is where a delicate but essential issue comes in. The expansion p(x, y) actually contains the (x 1/2) (x 1/2) series e− − which should be extracted. By dividing the series p(x, y) by the series e− − , we obtain the series p(s). A laborious calculation of p(s) gives the following fourth order expansion:

(x 1/2) p(x, y) = e− − p(s)

ip1 p2 ip3 p4 p(s) = p0 + + + + + ... y y2 y3 y4

p0 = 1 1 1 p = x(1 x) 1 −4 − 2 − 1 1 p = 5 + 3x + 6x2 + 2x3 3x4 2 24 4 − − ! 1 1 57x x2 9x4 p = 8 + + 8x3 + x5 x6 3 48 8 − 4 − 4 2 − − ! 1 29 49 25 13x 55x2 7 17x3 79x4 77x5 2x6 x8 p = × + + × + x7 + 4 96 2 − 960 − 8 2 12 − 8 − 10 − 3 4

y 1/2 s − Thus, when multiplying by this factor, the expression 2πe turns out to be:

1/2 x iy 1/2 x iy y − − (x 1/2) y − y − e− − = 2πe 2π 2πe

y iy iπ y iy iπ 1 y y − 4 − 4 i2π( 8 2π log 2πe ) The multiplication of 2πe by e also is simpliﬁed: 2πe e = e − y y We ﬁnd the expression of an anamorphosed y: ey = log 2π 2π e Considering the fractional part, it follows that: ey = [ey] + ey . Let α be the angle α = 2π( ey ). y 1/2 x iy− y 1/2 x − − (x 1/2) − iα The expression 2πe e− − therefore becomes: 2π e .

3.7.3. Expansion of Exponential Part q

  X ( )k+1( )2k 1 !1 2k !1 2k 1 2/y − B2k  ix0 − ix1 −  q(x, y) = ∞ −  1 + 1  k=1 2k(2k 1)  − y − y  − It is essential to calculate the polynomial q(x, y) = eiq(x,y) using the series expansion in series z z z2 z3 q e = 1 + 1! + 2! + 3! + ... for each of the terms of the sum. The laborious calculation of (s) using the exponential of Binet’s formula gives the order 6:

iq1 q2 iq3 q4 iq5 q6 q(s) = 1 + + + + + + + ... y y2 y3 y4 y5 y6

1 q = 1 3 1 1 q = (x + x ) 2 −18 − 6 0 1 31 1 q = x + 3x2 + x + 3x2 3 810 − 18 0 0 1 1 139 1 q = + 62x + 75x2 + 180x3 62x + 30x x + 75x2 + 180x3 4 −81 120 9 120 − 0 0 0 − 1 0 1 1 1 × × Information 2020, 11, 237 13 of 31

  43x2 x3 x2x 43x2 x x2 x3 9871 1 139x0 0 0 4 139x1 x0x1 0 1 1 0 1 1 4 q5 = +  + + x + + + + + x  36280 6− 810 − 60 2 0 − 810 18 6 − 60 6 2 1

1189x2 263x3 7x4 5x2x q = 2251 + 1 9871x0 + 0 + 0 0 x5 9871x1 + 67x0x1 0 1 6 − 36527 6 − 23357 23345 216 − 12 − 0 − 23357 540 − 72 x3x 1189x2 5x x2 x2x2 263x3 x x3 7x4 0 1 + 1 0 1 0 1 + 1 0 1 1 x5 − 6 23345 − 72 − 6 216 − 6 − 12 − 1 i 1 1 q(s) 1 + + x0 + x1 3y − 6y2 3 The polynomial q(s) is a symmetric polynomial at x = x and x = x 1. Its inverse is obtained 0 1 − by changing x to 1 x and y to y, (modulo a monomial in x1 = x 1/2)). − − /2 − 3.7.4. Final Expression of Ksi

y 1/2 x The expression of Ksi eventually becomes: ξ(s) = − eiα (s) 2π A (s) is a polynomial with complex coeﬃcients, which we calculate: A ia a ia (s) = p(s)q(s) = 1 + 1 + 2 + 3 + ... A y y2 y3

1 x x2 1 1 a = + = (x(1 x)) 1 12 − 2 2 12 − 2 − 2 1 x x2 x3 x4 1 1 + x + 3x a2 = + + = + x(1 x) − −288 − 24 12 12 − 8 −2 122 − 24 × 139 x 19x2 x3 5x4 x5 x6 a3 = + + 5 3!123 − 192 − 576 36 96 − 48 − 48 × 571 139x 67x2 101x3 7x4 91x5 x7 x8 a4 = + + + + −5 4!124 − 20736 12960 3456 − 768 − 2880 96 384 × Finally, taking into account all the elements, we can present the ﬁnal formula, by separating the inﬂuence of x as follows:

2i 2 4 139i 2 571 (s) = 1 + + × × A 24y − (24y)2 15(24y)3 − 15(24y)4

2 1 11x 2 3 4 x(1 x) i 1+x+3x 2 3 x +4x +2x + − + + i − − − y 2 − 24y 4 24y2 − × 2 3 ! 139 53x + 319x + 107x 5x4 5x5 x6 + − 54 − 90 30 15 − − − 16 24y3 ×

With the expression of this polynomial , we discover that it is possible to perform two digressive A 1 rotations. The ﬁrst is the rotation 3 of angle θ3 = 24y , and a fourth rotation 4 (which will not be R 2 − R (x 1/2) sensitive on the critical line of angle θ = − ). These rotations are digressive, because there is 4 − 2y a series as a remainder. With these rotations, all computations carried out, and the polynomial A ﬁnally becomes the polynomial : K Information 2020, 11, 237 14 of 31

(s) K 2 i i(x 1/2) 2 2 − i(x 1/2) (x 1/2)( 1+10x 8x ) = e− 24y − 2y + − + − − − y 48y2 7 5 23x2 3 4 5 i +(x 1/2) x+ x +x 2x + − 60 − − 12 − 6 − − 48y3 7 133 2753x 707x2 181x3 87x4 5 6 +(x 1/2) + + 5x +12x + − 120 − 720 360 60 − 10 − 10 − 242 y4

1 2 1 2 1 2 i i(x /2) y / x y / x − − The ﬁnal Ksi formula is therefore: ξ(s) = − eiα (s) = − eiα e˘ 24y e˘− 2y 2π K 2π

3.7.5. Special Case of Ksi: x = 1/2 We can resume the calculations of p, q, r = pq taking into account the angle of rotation θ = 1 , 3 − 24y in order to reveal the digressive rotations:

3i 32 139i 3 52 19 32851i 592153 3 6090313i 473920289 p = 1 + + × × + × + + ... − 8y − 2!(8y)2 3!(8y)3 4!(8y)4 − 5!(8y)5 − 6!(8y)6 7!(8y)7 8!(8y)8

2 (247+33)i (263 7+34) (2 32 132+35)i p = 1 3i 3 + + × × × − 8y − 2!(8y)2 3!(8y)3 4!(8y)4 − 5!(8y)5 6 4 7 (64 9241+3 ) (2 3 107 3557+3 )i 27379 9769+38 × + × × + × − 6!(8y)6 7!(8y)7 8!(8y)8 3i 3i 7 2i 7 12 1019i 9241 3 107 3557i 379 9769 p(y) = e˘ −8y = e −8y + × + × + × × + × 3!y3 4!y4 − 5!y5 − 6!y6 8 7!y7 2 8!y8 × × i 1 (73/10)i 131/5 (2921/14)i 97423/70 (481601/20)i 214939 q = 1 + + + + + ... 3y − 2!(3y)2 − 3!(3y)3 4!(3y)4 5!(3y)5 − 6!(3y)6 − 7!(3y)7 8!(3y)8 i i 7 9i 18 7 9 323i 97283 481561i 214181 q(y) = e˘ 3y = e 3y × + × + × + − 10 3!y3 5 4!y4 14 5!y5 − 70 6!y6 − 20 7!y7 8!y8 × × × × × 4 2 6 2 i 1 (2 3 7 5)i 26327 5 (2 3 29 10079+7)i r = pq = 1 − − × 24y 2!(24y)2 5 3!(24y)3 5 4!(24y)4 7 5!(24y)5 − − − × − × − × 253217807+35 (243243 11009087 5)i × − 35 6!(24y)6 5 7!(24y)7 − × − × 2732132131 23027 5 × − 5 8!(24y)8 − × The ﬁnal solution therefore provides: θ 2 17807θ 2i 29 10079θ 3 43 11009087θ 4i r iθ3 3 3 3 3 = e 5y 7i 7θ3 + 7 2! ×35 3! + × 35 4! − − × − × × 2 5 13 131 23027θ3 7× 5! ... − ×

1 1 1 iπ /2 /2 /2 y − iq(1/2,y) y − iα iα iθ ξ(𝓈) = e 4 p(1/2, y)e = e r(y) = e e˘ 3 2πe 2π × 2 7iθ 2 1/(7 24 ) In total: ξ( ) = eiαe˘iθ3 = ei(α+θ3) 1 3 1 + iθ + 17807/49 θ 2 × iθ 3 + ... − 5y 3 2! 3 − 3! 3 2 More approximately: ξ( ) = eiαe˘iθ3 ei(α+θ3) 1 7iε eiθ3 − 5y

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1 1 As ξ (s) = ξ(1 s), we can verify that if x = 1/2, ξ(𝓈) − = ξ( ) or ξ( )ξ( ) = 1. We verify − − numerically that r r = 1. × ( )i ( )i r r = (1 i 1 1003/5 4027/5 5128423/7 168359651/35 × − 24y − 2!(24y)2 − 3!(24y)3 − 4!(24y)4 − 5!(24y)5 − 6!(24y)6 (68168266699/5)i 587283555451/5 ) − 7!(24y)7 − 8!(24y)8 ( )i ( )i (1 + i 1 + 1003/5 4027/5 + 5128423/7 × 24y − 2!(24y)2 3!(24y)3 − 4!(24y)4 5!(24y)5 168359651/35 + (68168266699/5)i 587283555451/5 ) − 6!(24y)6 7!(24y)7 − 8!(24y)8

18 312603562112249 r r = 1 + × + ... × 7 10!(24y)10 × 4. Results This section first presents the computer results and the formula on the critical line and continues with a graphical presentation and a numerical calculation. Then, the section apprehends the calculations in the critical strip in order to prove that the points s of the critical line are located on the limit of a property of the functional relation. Finally, the section presents a geometric interpretation of the functional equation, which clarifies the RH. To understand the arrangements of both surfaces which contribute to the RH, the mechanisms of the joint undulations of the real and imaginary surfaces of the Zeta function and the Ksi function must be demystified. The local behavior of the Zeta and Ksi functions is mainly constructed in relation to the Gamma function. The Ksi function, which brings two symmetric values of the Zeta function into constant y, is essentially a ratio of two Gamma functions. It is therefore essential to analyze and visualize the behavior of the Gamma function and the Ksi function in the critical strip, and to focus attention on the specific case of the critical line. The essential constraint is the functional equation which imposes a strong requirement to constant y, between x and 1 x. − 4.1. Concordant Undulations of the Two Curves ζ( ) and Bézout’s Identity on the Critical Line

In this paragraph, we show that we can ﬁnd a function ϕ(y) ζ(1/2, y) that is canceled out over ◦ the entire longitudinal section of the critical line. This function depends on y only.

4.1.1. Result Statement On the critical line , there is a congruence, in a Bézout identity form, between the real part L (ζ(s)) = a and the imaginary part (ζ(s)) = b of the Zeta function. < =

y R ; x = 1/2; = 1/2 + i y; ζ( ) = a + ib; u(y), v(y) : a u(y) + b v(y) = 0 ∀ ∈ ∃ 4.1.2. Graphical Approach for the Function ϕ On the critical line, we first try to find a sequence of operations that allows us to graphically compare both curves (real part and imaginary part) of the Zeta function. Then, we deduce a linear combination of the real and imaginary parts that cancel out. We want to preserve the status of non-trivial zeros, which invites us to utilize linear combinations, in particular rotations. First rotation: approximation of the curves • The first rotation (Figure5) is obviously the rotation (θ ) = eiπ/4 that restores the symmetry R1 1 between the values of the real part and the imaginary part, by removing the initial bias of the two 1 1 cos(y log(1))+i sin y log(1) parts due to the first term of the Riemann sum ζ(s) = 1s + 2s + ... = 1x + ... = 1 1 (1 + 0 i) + s ... = 1 + 0 + s ... since cos(0) = 1 and sin(0) = 0. The real part and the imaginary × 2 2

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⁄ The first rotation (Figure 5) is obviously the rotation ℛ() = that restores the symmetry between the values of the real part and the imaginary part, by removing the initial bias of the two (()) () parts due to the first term of the Riemann sum () = + +⋯= +⋯= (Information1+0×2020) + , 11…=1+0+, 237 … since cos(0) = 1 and sin(0) = 0. The real part and the imaginary16 of 31 part of the new function are thus transformed, oscillate similarly around the value 0, one not being higherpart of thethan new the function other are(on thusaverage): transformed, obviously, oscillate the similarlynon-trivial around zeros the are value invariant0, one notby being this transformation.higher than the other (on average): obviously, the non-trivial zeros are invariant by this transformation.

Figure 5. Raw real (in blue) and imaginary (in red) curve ζ(s) versus y and after rotation of the axes by Figure 5. Raw real (in blue) and imaginary (in red) curve () versus and after rotation of the axes 45 . The barycenters of the two curves ζ(s)eiπ/4 versus y have moved closer together. by◦ 45°. The barycenters of the two curves ()⁄ versus have moved closer together. Second transformation: regularization of the axes of the curves and second rotation • Second transformation: regularization of the axes of the curves and second rotation The second transformation is more delicate and more involved. It plays, both globally on the spacingThe ofsecond zeros, transformation in order to obtain is amore uniform delicate distribution, and more and involved. locally toIt createplays, aboth local globally equivalence on the of spacingthe Zeta of function’s zeros, in behavior. order to obtain We therefore a uniform perform distribution, an anamorphosis and locally of theto createy axis a and local use equivalence the fractional of thepart Zeta of this function’s variable, behavior. in order toWe deduce therefore an angle, perform which an weanamorphosis use on the oneof the hand inaxis an and equivalence use the fractionalrelation and part on of the this other variable, hand inin a order second to rotation. deduce an angle, which we use on the one hand in an equivalenceIndeed, relation to homogenize and on thethe shape other ofhand the Zetain a second function, rotation. it is necessary to anamorphose the imaginary y y axis byIndeed, a stretching to homogenize transformation the shape of this of axisthe Zetaiy function,iey = it(log itt is 1necessary) = i 2π log to2π eanamorphose. This imaginary the → − imaginaryaxis change axis makes by a it stretching possible to transformation obtain a Zeta graph of this with axis periodic → = undulations(log − that 1) = are morelog constant. This in probability. The anamorphosis makes it possible to reveal an important property of the Zeta function imaginary axis change makes it possible to obtain a Zeta graph with periodic undulations that are = y y morevia the constant fractional in probability. part ey of The the variableanamorphosisey 2 πmakelog s2 πite possible. Specifically, to reveal the an following important relationships property of are observed: the Zeta function via the fractional part {} of the variable = log . Specifically, the following relationships are observed: 𝓈 = 1/2 + iy : ey = 1/8 = 𝒷 0; a/ 48y ; ey = 5/8 = a 0; /a 48y ∀ ⇒ ⇒ ∀ = ½ + ∶ {} =⅛ ⟹ ≅0;⁄ ≅ 48; {} =⅝ ⟹ ≅0;⁄ ≅ 48 = 1/2 + iy : ey = 3/8 = a = ; ey = 7/8 = a = ∀ ⇒ − ⇒ The results for ey ∀= k =/8; ½k += 0, ∶ { 1,} 2,= 3,⅜ 4,⟹ 5, 6,=− 7 are; summarized{} = ⅞ ⟹ in Table= 2. { } TheTable results 2. Index, for fractional {} =⁄ part 8; of = the {0, anamorphosed 1, 2, 3, 4, 5, 6, 7} variable, are summarized tangent of the in halfTable angle, 2. ratio % of the imaginary part (ζ(1/2+iy)) to the real part (ζ(1/2+iy)). This is a major property of the zeta function. = < TableFrom this2. Index, property, fractional an equivalence part of the relation anamorphosed can be defined. variable, tangent of the half angle, ratio of the imaginary part ℑ(½+) to the real part ℜ(½+). This is a major property of the zeta ~ 1 function. From kthis property,{y an} equivalenceα relation cantan( beα defined./2) % (resp. %− ) 0 0 π/4 √2 1 √2 1 k {} (/) − (resp. ) − 1 1/8 0 0 % 1 = 48y 0 0 4⁄ √2 −1 √ 2 −1 − − 2 1/4 7π/4 √2 1 √2 1 1 ⅛ 0 0 − − = −48 − − 3 3/8 3π/2 1 1 2 ¼ 7⁄ 4 −(√2 −1) − −(√2 −1) − 4 1/2 5π/4 √2 + 1 √2 + 1 ⅜ 3⁄ 2 −1 − −1 − 5 3 5/8 π % = 48y ∞ 6 4 3/4 ½ 5⁄3π 4 /4−(√2 +1)√ 2 +−(1 √2 +1) √2 + 1 7 5 7/8 ⅝ π/2∞ 1 = 48 1 6 ¾ 3⁄ 4 √2 +1 √2 +1 7 ⅞ 2⁄ 1 1 (ζ(1/2+iy)) α The ratio is in fact the tangent of the half angle (Figure6a): % = = tan . The imaginary (ζ(1/2+iy)) 2 < axis anamorphosis thus makes it possible to define an equivalence relation (Figure6b) in order to reduce the Zeta function study of y from 0 to infinity, to an examination in the quotient space (Figure7),

1 2 Information 2020, 11, x FOR PEER REVIEW 17 of 32 Information 2020, 11, x FOR PEER REVIEW 17 of 32

Information 2020, 11, x FOR PEER REVIEW ℑ(½) 17 of 32 The ratio is in fact the tangent of the half angle (Figure 6a): = ℑ(½≅tan) . The The ratio is in fact the tangent of the half angle (Figure 6a): =ℜ(½) ≅tan . The ℑℜ(½(½) ) imaginaryThe ratio axis anamorphosisis in fact the thustangent makes of itthe possible half angle to define (Figure an equivalence 6a): = relation≅tan (Figure 6b). The in imaginary axis anamorphosis thus makes it possible to define an equivalenceℜ(½ relation) (Figure 6b) in order to reduce the Zeta function study of from 0 to infinity, to an examination in the quotient Informationimaginaryorder to2020 axisreduce, 11 ,anamorphosis 237 the Zeta function thus makes study it of possible from to 0 define to infinity, an equivalence to an examination relation (Figure in the 17quotient6b) of in 31 space (Figure 7), that is to say over a finite interval {} =[0,1). The ratio =⁄ = orderspace to reduce(Figure the 7), Zeta that function is to studysay over of afrom finite 0 tointerval infinity, {to} an=[0,1) examination. The ratioin the =quotient⁄ = ℑ(½+)/ℜ(½+) is constant (to within epsilon) for each equivalence class ({}). spaceℑ (½+(Figure) /ℜ7), that(½+ is )to issay constant over (toa withinfinite epsilointervaln) for{ each} =[0,1) equivalence. The ratioclass (={}).⁄ = 1 1 thatℑ( is½+ to say)/ℜ over( a½+ ﬁnite) interval is constant ey = (to[0, within 1). The epsilo ration)% =for 𝒷 each/ a = equivalence(ζ( /2+iy class))/ ((ζ({/}2).+ iy)) is ∀ = ½ + ; = −2 {};() =(½+ ) = += ⟹ = ≅tan < constant (to within∀ epsilon) = ½ + ; for each =4 equivalence−2{};( class) =(½+( ey ). ) = + ⟹ = ≅tan2 4 C 2 ∀ = ½ + ; = −2{};() =(½+) = + ⟹ = ≅tan π 4 2α 𝓈 = 1/2 + iy; α = 2π ey ; ζ( ) = ζ(1/2+iy) = a + i = % = tan ∀ 4 − ⇒ a 2

(a) (b) (a) (b) Figure 6. (a) tan ⁄ 2 in blue and(a) ratio in red versus axis: Both curves(b) are superimposed, but the Figure 6. (a) tan ⁄ 2 in blue and ratio in red versus axis: Both curves are superimposed, but the reader observes only one curve due to their similarity. (b) Zeta function versus {} +⅛: In blue the Figurereader 6. (observesa) tantanα /⁄ 2 only in in blue blueone andcurvand %e ratiodue ratio to in in their red red versus similarity.versusy . axis: axis:(b) BothZeta Both curvesfunction curves are ar versuse superimposed,superimposed, {} +⅛: In butbut blue the the real part, in red the imaginary part for four different segments. readerreal observespart,observes in red only the one imaginary curvecurve due part toto for theirtheir four similarity.similarity. different ((segments.bb)) ZetaZeta functionfunction versusversus {ey}++⅛1/8:: InIn blueblue thethe real part,part, inin redred thethe imaginaryimaginary partpart forfor fourfour didifferentﬀerent segments.segments.

Figure 7. Three consecutive classes of equivalence of the zeta function for various intervals: in blue FigureFigure 7. Three7. Three consecutive consecutive classes classes of of equivalence equivalence of of the the zeta zeta function function for for various variousy intervals: intervals: in bluein blue the real part, in red the imaginary part of several superposed intervals of zeta curves. theFigurethe real real 7. part, Three part, in redconsecutivein red the the imaginary imaginary classes part of part equivalence of severalof several superposed of superposed the zeta function intervals intervals for of zetavariousof zeta curves. curves. intervals: in blue the real part, in red the imaginary part of several superposed intervals of zeta curves. The second rotation ℛ() is decisive (Figure 8). It is based on the use of the fractional part TheThe second second rotation rotation2 (ℛθ2)(is) decisive is decisive (Figure (Figure8). It 8). is basedIt is based on the on use the of use the of fractional the fractional part eypart. R {}. This variable, which varies( between) 0 and 1, multiplied by 2, is interpreted as an angle= y= This{} variable,The. This second variable, which rotation varieswhich ℛ betweenvaries between is 0decisiveand 1 ,0 multiplied (Figureand 1, multiplied8). by It 2isπ ,based is interpretedby on2 ,the is interpreted use as anof anglethe fractionalasθ an2 angle2 πpart e. = −2{} . Taking into account the first rotation ℛ , we consider the sum angle = −+= Taking{}−2. This{ into }variable, account which the ﬁrstvaries rotation between 10, weand consider 1, multipliedℛ the sum by angle2, is αinterpreted= θ1 + θ 2as= anπ =angle/4 2 π+ey=. = . Taking into account the Rfirst rotation , we consider the sum angle − 4−⁄ 2{}. This rotation is justified by the fact that the ratio ℜ()ℑ 1 () for =½ is close This−24−⁄{ rotation} . 2Taking{ is}. justiﬁedThis into rotation account by the is fact justifiedthe that first the byrotation ratio the fact (ζℛ that(s)), we /the ( ζconsiderratio(s)) forℜ xthe(=) ℑ/sum2 is close(angle) tofor the= =½ tangent + is close of= to the tangent of a half angle, which is none other< than the= angle . This angle structures the Zeta a4−⁄ halfto the angle,2 tangent{}.which This of rotation a is half none angle, otheris justified which than the byis nonethe angle fact otherα .that This than the angle the ratio angleα structuresℜ (. )Thisℑ the angle( Zeta) for function structures =½ behavior is the close Zeta function behavior on the critical line. Non-trivial zeros are always preserved here. onto functionthe the tangent critical behavior line.of a half Non-trivial on angle, the critical which zeros line. is are none Non-trivial always other preserved than zeros the are here.angle always . This preserved angle here. structures the Zeta function behavior on the critical line. Non-trivial zeros are always preserved here.

Figure 8. ζ(s)eiα and congruence au + bv, after the second pure rotation. Figure 8. () and congruence + , after the second pure rotation. Figure 8. () and congruence + , after the second pure rotation. It is in this sense that the() spirit of this presentation + can be brought closer to the architecture of Figure 8. and congruence , after the second pure rotation. Deligne’s RH demonstration by algebraic geometry for ﬁnite sets (curves of genus g). This somewhat daring suggestion is due to the fact that the variable y/2π log(y/2πe)mod 1 produces, by this folding, an equivalence relation; a unique scheme for studying the Zeta function over a ﬁnite interval (the

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It is in this sense that the spirit of this presentation can be brought closer to the architecture of Deligne’sInformation 2020 RH, demonstration11, 237 by algebraic geometry for finite sets (curves of genus ). This somewhat18 of 31 daring suggestion is due to the fact that the variable 2⁄ log(2⁄ ) 1 produces, by this folding, an equivalence relation; a unique scheme for studying the Zeta function over a finite interval (thequotient quotient space), space), which which makes makes it possible it possible to analyze to anal itsyze properties its properties over thisover interval this interval rather rather than along than alongthe entire the entire critical critical line. line. • Third rotation: juxtaposition juxtaposition of of curves, curves, erosion erosion of of the the deviation deviation from from zero zero crossings of the real • and imaginary parts. The third rotation ℛ brings us closer to our target of 0. It corrects the previous result by taking into accountThe third the rotation essential3 bringsproperty us closerof the toZeta our function target of at0. It=½ corrects of thewhich previous the real result part by and taking the R imaginaryinto account part the cancel essential each propertyother out, ofand the this Zeta alternately function within at x = the1/ 2vicinityof which of =2×⅝ the real part and and = the 2imaginary × ⅛, respectively. part cancel It each uses other an angle out, and that this corresponds alternately to within the remarkable the vicinity ratio of α =of 2≅−π 5/8 andfor × !1 α = 2π 1/8, respectively. It uses an angle that corresponds to the remarkable ratio of α for ℜ()ℑ× () and ≅ for ℑ()ℜ () for the values ⅛ and ⅝., respectively. We− 4! ythus !1 (ζ(s))/ (ζ(s)) and α 4!y for (ζ(s))/ (ζ(s)) for the values and ., respectively. We thus see see< emerging= a rotation angle == − 1⁄ 24<. However, this rotation is not pure, it is digressive. To emerging a rotation angle θ3 = 1/24y. However, this rotation is not pure, it is digressive. To finally − −1⁄ 24 finallyreach our reach target our of target zero, of a finalzero, power a final series power expansion series expansion in 1/24 iny allows us toallows conclude us to byconclude numerical by − numericalfine-tuning fine-tuning (Figure9). (Figure 9).

Figure 9. Congruence au + bv, after the third digressive rotation. The third rotation ﬁne-tuning is of Figure 9. Congruence12 + , after the third digressive rotation. The third rotation fine-tuning is of the order of 10− . the order of 10−12. We then graphically deduce a remarkable identity of equality between two products of two factors, the ﬁrstWe factorsthen graphically of each member deduce being a remarkable the real and identi imaginaryty of equality parts of between the Zeta two function, products the secondof two factors,factors dependingthe first factors on the of each trigonometric member being functions the real of theand angle imaginaryα, supplemented parts of the byZeta a powerfunction, series the secondexpansion factors dependent depending on 1/ ony. Thethe trigonometric nullity of a member functions leads of to the the angle nullity of, supplemented the second member, by a power that is seriesto say expansion the nullity ofdependent a factor ofon this 1/ second. The membernullity of (or a both).member These leads second to the factors nullity in of each the member second member,cancel each that other is to out say as the a functionnullity of of a thefactor angle of thα,is but second are generally member non-zero.(or both). These The general second relation factors onin eachthe critical member line cancel is as follows: each other out as a function of the angle , but are generally non-zero. The general relation on the critical line is as follows: y y 𝓈 = 1/2 + iy C ; ey = log ; α = 2π(1/8 ey ); α1 α2 ye1 = ye2 ∀ ∀ = ½ + ∈ ∈ L ℒ ⊂ ⊂ ℂ ; =2π log 2πe ; = 2(⅛−− {}); ∼∼⇐⇒⟺ { } = { } 2 2 ! 1 1 1 ε(y) = 1 + 1 + ... + o 1 ; () =24y +413.48y3 +⋯+y2n+1 ; 24 413.48 ! 1 1 1 (y) = + ... + o 1 2 1 4 2n1 () =2(24y) +9886.5y …+y 2(24) 9886.5 (sin α ε(y)) (ζ(s)) (1 + cos α (y)) (ζ(s)) = 0 − < − − = (sin − ())ℜ() − 1 + cos − ()ℑ()= 0 4.1.3. Numerical Approach for the Function ϕ 4.1.3.It Numerical remains to Approach rationalize for these the Function results by power series expansions of the functional relation ξ(s) and it remains to be understood the zeroing and synchronization by the undulations of the Gamma It remains to rationalize these results by power series expansions of the functional relation () function and the Ksi function. The Riemann functional relationship makes it possible to analyze the and it remains to be understood the zeroing and synchronization by the undulations of the Gamma result found on the critical line. Indeed for x = 1/2, the functional relation is written: function and the Ksi function. The Riemann functional relationship makes it possible to analyze the

=½ result found on the critical line. Indeed for Γ /, 2the functional relation is written: ζ( ) ζ ( ) 𝒾 𝓎 ξ( ) = = = π ; = 1/2 + ; = 1/2 ; ζ( ˆ ) ζ Γ( /2) −

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Since Γ(z) = Γ(z) and ζ(z) = ζ(z), it follows that ||ξ(s) = 1||, and this function ξ(s) is then:

a + ib ξ(𝓈) = = eiω a 1 eiω + ib 1 + eiω = 0 a ib ⇒ × − × − Hence: au + bv = 0.

u = (1 cos(ω)) ; v = sin(ω) or u = sin(ω); v = 1 + cos(ω) − − − The pair (u, v) is therefore a morphism ϕ whose kernel Ker(ϕ) is the critical line . This relation L (ζ(s)) (1 cos(ω)) (ζ(s)) sin(ω) = 0, derived exclusively from the functional equation, < × − − = × which indicates that the critical line is a fertile domain with non-trivial zeros, justifies the adverb “presumably” (wahrscheinlich) of Riemann’s manuscript, at the origin of the conjecture. Now we need to specify the angle ω. We therefore operate three rotations of the Zeta function on the critical line. The first rotation 1 modifies the axes of the complex coordinate system C, transformed by the R π diagonals of C. The term 4 follows in fact from the first term in the Riemann series:

s cos(y log(1)) i sin(y log(1)) 1− = = cos 0 i sin 0 = 1 1x − 1x − This first term in the series introduces an imbalance between the real part and the imaginary part, which generally places the real surface (ζ(s)) above the imaginary surface (ζ(s)). This imbalance < = is obviously perpetuated in all the Zeta function formulas. This is why the coefficients 1/8 or 7/8 are often found in the Zeta function formulas in scientific papers. A rotation ζ(s)eiπ/4 in the complex space makes it possible to remove this asymmetry, without changing the position of non-trivial zeros. To specify the second rotation , we must first modify the scale of the y axis, in order to obtain R2 a uniform distribution of non-trivial zeros. This anamorphosis (elongation of the axis) also makes it possible to homogenize the search for non-trivial zeros on a contiguous sequence of intervals In of the whole critical line. The congruence relation involves ey, the anamorphosed variable y, which is the integral of the probability density of non-trivial zeros on the line x = 1/2: Z y t dt y y ey = log = log = t(log t 1) 0 2π 2π 2π 2πe − From this variable ey, we determine its fractional part ey : ey = [ey] + ey . This variable ey [0, 1) ∈ determines an equivalence relation on the iy axis of C: y1 y2 ye1 = ye2 . ∼ ⇐⇒ The fractional part allows to introduce an angle θ2 [0, 2π): θ2 = 2π ey . The first two rotations ∈ − thus define an angle α: α = 2π(1/8 ey ) = (π/4 2π ey )mod(2π). The angle α determines the same − − equivalence relation over the interval [0, 2π). In Section3 on preparatory calculations, we developed the Ksi function. More precisely:   3i 1 7 8i 21 8 4 1019i 2 9241 380599i 3702451 −8y   p = e +  × + × × × + + + ... 3 (8y)3 (8y)4 − 5(8y)5 − 75(8y)6 35(8y)7 105(8y)8    i 1 70i 35/2 14 323i 70 10817 20 53509i 11941/2 3y   q = e  + + × × × + + ... − 93!(3y)3 4!(3y)4 5!(3y)5 − 6!(3y)6 − 7!(3y)7 8!(3y)8 

i 1 ((7 3!24 5)/5)i (7 4!24 5)/5 r = 1 × − × − − 24y − 2!(24y)2 − 3!(24y)3 − 4!(24y)4 ((263229 10079+7)/7)i (253217807+35)/35 × − 5!(24y)5 − 6!(24y)6 ((243243 11009087 5)/5)i (2732132131 23027 5)/5 × − × − + ... − 7!(24y)7 − 8!(24y)8

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However, at this stage, we observe that a new rotation takes shape under this series r. We must R3 ﬁlter r in order to extract the rotation θ = 1 by dividing r by the series eiθ3 : 3 − 24y z2 z3 z4 z5 z6 z7 z8 i ez = 1 + z + + + + + + + ... ; z = iθ = 2 6 24 120 720 5040 40320 3 −24y

The division provides a remainder since it is a digressive rotation: The result r then becomes:

3 2 3 iθ iθ 24iθ3 2 2 5 17807θ3 29 10079iθ3 r = e˘ 3 = e 3 + 5 7 + 5 7 iθ + × + × 527 × × 3 2! 3! 4 2 5 43 11009087θ3 5 13 131 23027iθ3 ℴ 9 + × 4! + × × 5!× + 1/y

i 3 iθ3 iθ3 7 24 θ3 4 iθ3 3 r = e˘ = e + × (1 + iθ3) + 1/y e 1 + i33.6 θ3 5 ≈ × We ﬁnally get the result: iω iα iα i(θ1+θ2) iθ3 iα iθ3 i(α+β) i(α+θ ) ξ(𝓈) = e = e p(y)q(y) = e r(y) = e e + z˘3(y) = e e˘ = e = e L

Approximately, at fourth order, we can use, with a small error, the formula: ξ( ) = eiα eiθ3 + i 33.6 θ 3e˘iθ3 = ei(α+θ3) 1 + i 33.6 θ 3 + 1/y5 × 3 3 It goes without saying that, when a non-trivial zero occurs on the critical line, we are in the presence of an equation ζ( ) = ξ( )ζ(1 ) = ξ( )ζ( ), which is written as 0 = ξ( ) 0. As the − × Zeta function is holomorphic, L’Hôpital’s rule applies and ζ0( ) = ξ( )ζ0( ), which can be veriﬁed numerically. The summary of the formulas is as follows on the critical line : L

X !k i/24y i/24y 1 i iθ e˘− = ( ) = e− + 𝓀 (y) = ∞ − + (y) = e K k=0 k! 24y L

β = arg( ( )) = θ = θ˘3 = θ3 + θ(y) = 1/24y + θ(y) ; ( ) = 1. K L − K 𝜃 17807 × 12𝑖 4 × 2487259 11 × 41 × 75654497𝑖 𝓀𝑦 = 7 × 24𝑖 − 7 × 24𝜃 + 𝜃 − 𝜃 + 𝜃 5 7 5×7×9 8×5×7×9 47 × 108544391 1 1 + 𝜃 + ℴ ; 𝜃 =− 5×7×9 𝑦 24𝑦

InformationWith 2020 this, 11 formula,, x FOR PEER the REVIEW gap between ξ(s) and the estimation ξ∗(s) is negligible (Figure 10). 21 of 32

FigureFigure 10. 10.The Theξ (s)()ξ−(s∗)()error error (real (real and and imaginary imaginary parts) parts) versus versus the they axis axisis of is the oforder the order of 10 of15 . − ∗ − 10−15.

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4.2. Unsynchronized Undulations in s and (1 s) of Surfaces ζ(s) in the Critical Strip − Just as we investigated the Zeta function by a longitudinal section along the critical line, we now examine the Zeta function along a section x = xp. Of course, because of the functional equation, we are trying to ﬁnd a Bézout identity function of xp and 1 xp, but this search turns out to be in vain, because − of the function ξ, which is essentially a Gamma ratio that makes it impossible to discover a morphism ϕ xp, y , that is to say a pair of functions (u, v) which cancels out the expression au + bv, and the pair associated with functions (uˆ, vˆ), which cancels out the expression aˆuˆ + bˆvˆ.

4.2.1. Result Statement In the critical strip , there are obviously classes of congruence between the real part (ζ(s)) = a S < and the imaginary part (ζ(s)) = b of the Zeta function. = y R ; x [0, 1]; s = x + i y; ζ(s) = a + ib; u, v, c : a u(x, y) + b v(x, y) = c(x, y) ∀ ∈ ∈ ∃ × × However, in the critical strip , there is no congruence, in a Bézout identity form, between the S\L real part (ζ(s)) = a and the imaginary part (ζ(s)) = b of the Zeta function, with a relation which < = involves two pairs (u, v), (uˆ, vˆ), relative to x and 1 x, functions of only the coordinates (x, y). − X s ζ(s) = ∞ n− = (ζ(s)) + i (ζ(s)); (ζ(s)) = a(x, y); (ζ(s)) = b(x, y). n=1 < = < =

y R; x [0, 1], x , 1/2; s = x + i y; s , 1; ζ(s) = a + ib; ζ(sˆ) = aˆ + ibˆ; ∀ ∈ ∀ ∈ @(u, v), (uˆ, vˆ) : a u(x, y) + b v(x, y) = aˆ uˆ(1 x, y) + bˆ vˆ(1 x, y) = 0. − − − − 4.2.2. Graphical Approach for the Function ϕ With the results on the critical line, it is natural to graphically visualize the Zeta surfaces after the three rotations that we performed in the above paragraph. We do not make the assumption of x being small compared to y, and we execute the calculations and the surface ﬁtting on the Ksi raw values. The original Ksi function and the parameter estimates are presented in Figure 11. ! x1 i(α+θ ) a1i a2 a3i a4 ξ(s) = t− /2 e 3 1 + + + + y y2 y3 y4 a = 1 x2 x + 1 = 1 x(1 x) 1 ; a is symmetrical in x , x . 1 2 − 4 − 2 − − 4 1 0 1 1 2 1 1 1 1 10 7 7 1 a = x4 + x3 + x2 x = x 4 + x 3 x 2 + x 2 8 − 3 2 − 6 − 16 8 − 1 3 1 − 2 1 6 1 − 16

a2 is not symmetrical in x0, x1

1 1 1 1 1 1 1 a = x6 x5 + x4 + x3 x2 x + 3 −48 − 48 21 32 − 33.4 − 153.5 407.4 1 1 1 1 1 1 1 a = x6 x5 + x4 + x3 + x2 x 4 17.6 − 8.5 19.5 202 94.3 − 148 − 3000

The graphical examination shows that a2 is not symmetrical in x0 and x1. It is therefore impossible to ﬁnd a Bézout identity type in the critical strip, as on the critical line. Information 2020, 11, 237 22 of 31 Information 2020, 11, x FOR PEER REVIEW 23 of 32

Figure 11. Above: The Ksi original function, real part and imaginary part. Below: ∗ and ∗. The Figure 11. Above: The Ksi original function, real part and imaginary part. Below: a2∗ and a1∗ . ∗ Thedrawing drawing of the of real the realpart part seemsa2∗ seems to strengthen to strengthen the the RH. RH. Indeed, Indeed, the the plot plot shows shows that that the the remainder remainder is notis not symmetrical symmetrical with with respect respect to to=½x = 1,/ and2, and that that for for =½x = 1 / the2 the remainder remainder is isequal equal to to0. 0.

4.2.3. Numerical Approach for the Function ϕ At thisthis stage,stage, we we can can resume resume the the calculations calculations and reviewand review the formalization the formalization using the using Zeta the function Zeta propertiesfunction properties in the critical in the strip. critical The strip. Ksifunction The Ksi numericalfunction numerical study helps study to lifthelps the to veil lift on the the veil previous on the previouscalculations. calculations. However, However, the divisibility the divisibility of the real of and the imaginaryreal and imaginary parts of ζ (partss) is moreof ( di) ﬃiscult more to difficultimplement, to implement, because we because add a further we add obstacle a further and anobstacle additional and an degree additional of freedom degree with of thefreedom emergence with of the symmetrical value ζ(1 s) in the formulas. The Riemann functional equation is a symmetry the emergence of the symmetrical− value (1−) in the formulas. The Riemann functional equation relation around the axis x = 1/2 between s and (1 s) in the critical strip . The Ksi function includes is a symmetry relation around the axis =½ between− and (1 − ) inS the critical strip . The Ksi in its internal structure a mechanism for taking into account the local undulations of both real and function includes in its internal structure a mechanism for taking into account the local undulations imaginary surfaces. This essential engine is driven by the fractional part ey of the anamorphosed{ } of both real and imaginaryy y surfaces. This essential engine is driven by the fractional part of the = = π variableanamorphosed of y,ey variable2π log of2π e,, thanks = tolog the angle, thanksα to 4the 2angleπ ey . = −2{}. − • ConstraintConstraint byby thethe functionalfunctional equationequation • ≔ { (, )} It is natural to look for the condition that if the point = ( ) is a non-trivial zero, It is natural to look for the condition that if the point M : s or x, y is a non-trivial zero, then ≔{̂ =1− (1−,−)} , ≔ {̅ =− (, − )} and n ≔̂̅ =1−+ (1−o then Mˆ := sˆ = 1 s or (1 x, y) , M := s = x iy or (x, y) and Mˆ := sˆ = 1 x + iy or (1 x, y) , ) are zeros also.− The pair− −(, ) that we are− looking for− must therefore contain− this requirement− are zeros also. The pair (u, v) that we are looking for must therefore contain this requirement and and exhibit this symmetry. It is thus advisable to consider = and = 1 − , given the exhibit this symmetry. It is thus advisable to consider x = xp and x = 1 xp, given the functional functional relation symmetry. Conceived with this constraint, the problem− becomes for all points relation symmetry. Conceived with this constraint, the problem becomes for all points sp = xp + iy of = + of these lines = to find pairs (,) such that: these lines x = xp to ﬁnd pairs up, vp such that: ∀ : ℜ () +ℑ () =0. y : ζ sp up(y) + ζ sp vp(y) = 0. ∀ < = ∀ : ℜ 1 − () +ℑ1− () =0 y : ζ 1 sp uˆp(y) + ζ 1 sp vˆp(y) = 0 ∀ < − = − , is the symmetric pair, associated with the pair ,. First, we calculate , as a function of ,. We assume that there is a pair (, ) such that + = 0. By symmetry, there

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uˆp, vˆp is the symmetric pair, associated with the pair up, vp . First, we calculate uˆp, vˆp as a function of up, vp . We assume that there is a pair (u, v) such that au + bv = 0. By symmetry, there exists a pair (uˆ, vˆ) such that aˆuˆ + bˆvˆ = 0. With the functional equation a + ib = (c + id) aˆ + ibˆ , it follows easily that: uˆ = cu + dv; vˆ = du + cv. The functional equation, in the critical strip, can be written as: − ζ(s) a + ib a + ib ξ(s) = = ξ(s) eiω = = ζ(sˆ) aˆ + ibˆ ka + ihb a + ib = (c + id) aˆ + ibˆ = (c + id)(ka + ihb); a, b, c, d, aˆ, bˆ, k, h R ∈ By separating the real and imaginary parts, we get two equations of type au + bv = 0:

a(1 kc) + b (hd) = 0 − a( kd) + b(1 hc) = 0 − − We then obtain a first constraint: hk c2 + d2 c(h + k) + 1 = 0. It involves the sum and the − product of h and k and the Ksi value. The constraint indicates that h and k are the real roots of the equation X2 (h + k)X + hk = 0. On the critical line, the values of k and h are k = 1; h = 1. Outside − − the critical line, obviously, we find a relation au + bv = 0, of which u and v intrinsically depend on both the real part and the imaginary part of ξ(s) and therefore the pair (a, b). For a given y, the pair (u, v) therefore depends on ζ(s), which is not suitable. It then remains to focus on the Ksi function to pull apart its mechanisms and get rid of the global engine, specific to the Zeta function, and to keep only the local constraints linked to the coordinates. Serial expansion • Calculation extensions, on the whole critical strip, are performed by deepening the functional relation. Ksi expansion in x, y, are achieved taking into account x0, x1/2 and x1. The operations lead to the third order formula of the Ksi function in the critical strip:  2 3 ! 2 x1 1 + x1 /2 4x1 3x1  ix1/2 /2 /2 /2 /2 1  iα x1/2 iα iθ3 − −  ξ(s) = ke (s) = t− e e + + + ℴ  K  2y 24y2 y3 

y π y y 1 t = ; x1 = x 1/2; α = 2π log ; θ = 2π /2 − 4 − 2π 2πe 3 −24y !! 1 x1/2 iα iθ3 ξ(s) t− e e + modulo x1 ≡ y3 /2 The above formulas contain rich information. The Ksi function is, within a similarity, the superposition of two fundamental series for the structuring of the Zeta function. The ﬁrst series θ is a rotation: h 3i !k X i i/(24y) ∞ 1 f (θ3) = e− = − k=0 k! 24y

The second series θ generates symmetry with respect to the axis x = 1/2: h 4i !k X ip Xk+1 ∞ k m f (θ4) = x1 2 ; pk = amx1 2 / k=1 y m=0 /

We see dawning in this remainder an underlying series composed of a new angle rotation 2 (x 1/2) θ = − . This rotation was obviously inoperative on the critical line, which is why it was 4 − 2y

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invisible,Information 2020 but, it11 must, x FOR be PEER considered REVIEW in the critical strip to completely filter the function Ksi from25 of its 32 Information 2020, 11, x FOR PEER REVIEW P n 25 of 32 influence of the reference series wn of sum function w = ∞ an(x 1/2) . This series which has h i n=0 − a very(, ) weak. The residue influence ( is) more was dicalculatedfficult to in detect Section graphically. 3. Figure The13 shows congruent the error Ksi estimation function, within of the theKsi (, ). The residue () was calculated in Section 3. Figure 13 shows the error estimation of the Ksi homothetyInformationfunction, 2020 which and, 11 the, xdecreases FOR four PEER rotations, asREVIEW grows. is exposed Figure in Figure14 shows 12. Hence,the modulus it is necessary of () to. consider (s) as25 twoof 32 function, which decreases as grows.iθ Figure+iθ 14 shows the modulus of (). K digressive rotations θ˘3 and θ˘4: (s) = e 3 4 + 𝓀 (s) with arg( (s)) = θ (x, y) = θ3 + θ4 + θ(x, y). K K S The(, residue ). The residue(s) was calculated() was calculated in Section in3. Section Figure 133. Figureshows 13 the shows error estimationthe error estimation of the Ksi of function, the Ksi whichfunction, decreases which decreases as y grows. as Figure grows. 14 shows Figure the 14 modulusshows the of modulus(s). of (). K

Figure 12. The congruent Ksi function, after homothety and pure rotations, for real part and Figure 12. The congruent Ksi function, after homothety and pure rotations, for real part and imaginary part. Again, the drawing of the real pa rt seems to strengthen the RH. Indeed, the plot showsimaginary that the part. remainder Again, the is drawingnot symmetrical of the real with part respect seems toto strengthen=½, and the that RH. for Indeed, =½ the the plot Figure 12. The congruent Ksi function, after homothety and pure rotations,=½ for real part and imaginary=½ Figureremaindershows 12. that isThe equal the congruent toremainder 0. Ksi is function, not symmetrical after homothety with respect and pureto rotations,, and forthat real for part and the part. Again, the drawing of the real part seems to strengthen the RH. Indeed, the plot shows that the imaginaryremainder part. is equalAgain, to the0. drawing of the real part seems to strengthen the RH. Indeed, the plot remainder is not symmetrical with respect to x = 1/2, and that for x = 1/2 the remainder is equal to 0. shows that the remainder is not symmetrical with respect to =½, and that for =½ the remainder is equal to 0.

∗ Figure 13. The error estimate () − (), real part and imaginary part are quite small ≈1011 . Figure 13. The error estimate ξ(s) ξ∗(s∗), real part and imaginary part are quite small 10− . Figure 13. The error estimate (−) − (), real part and imaginary part are quite small≈ ≈10 .

Figure 13. The error estimate () −∗(), real part and imaginary part are quite small ≈10.

FigureFigure 14. 14. ModulusModulus of of ((s)).. | (s))| ==00 on onthe thecritical criticalline line ℒonly. only. Figure 14. Modulus ofK ()K. |()| =0 on the critical lineL ℒ only. θ(,(x, y )) involvesinvolvesarg ((()s) , which), which is notis not symmetrical symmetrical in inx, 1, x1− . Consequently,. Consequently,β depends depends on (x, ony) (, ) involves (()), which is not symmetrical in1 − , 1−. Consequently,½ depends on and(, ) is notand symmetrical is not Figuresymmetrical in x14., 1 Modulusx. Thein of Ksi , 1− function(). |. (The)|ξ(=0s )Ksi= on π functionsthe/ 2criticalΓ(sˆ/ 2 )line/(Γ) (ℒs=/ only.2 ) is independentΓ(̂⁄)2 ⁄Γ(2⁄) of the is − − ( ) ½ ( ⁄)⁄ ( ⁄) independent(, ) and ofis the not Zeta symmetrical function. We in thus , found1−. theThe family Ksi offunction pairs (, ) and= (,)Γ independent̂ 2 Γ 2 is of independentthe(, Zeta ) involvesfunction, of the Zetain an(() function. irre),ducible which We is seriesthus not symmetricalfound form. theUnfortunately, family in , of1− pairs these. Consequently, (, )pairs and do ( ,not ) depends respectindependent theon (,symmetryof ) the and Zeta betweenis function,not symmetrical and in (1an −irre )in.ducible As this, 1− seriescondition. Theform. wasKsi Unfortunately, compulsory,function ( thethese) = \ℒ pairs½ domainΓ (dô⁄) 2not ⁄isΓ therefore(respect2⁄) is the independentsterilesymmetry and does ofbetween thenot Zeta admit function.and non-trivial (1 − We ). Aszeros.thus this found condition the family was compulsory,of pairs (, ) the and \ℒ ( , domain) independent is therefore sterile and does not admit non-trivial zeros. of the Zeta function, in an irreducible series form. Unfortunately, these pairs do not respect the symmetry between and (1 − ). As this condition was compulsory, the \ℒ domain is therefore sterile and does not admit non-trivial zeros. 5

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Zeta function. We thus found the family of pairs (u, v) and (uˆ, vˆ) independent of the Zeta function, in an irreducible series form. Unfortunately, these pairs do not respect the symmetry between s and (1 s). As this condition was compulsory, the domain is therefore sterile and does not admit − S\L non-trivial zeros. We observe the ﬁltered relation of the congruence of the series between s = x + iy and i K 𝓈 = 1/2 + iy: s : (s) modulo(x 1/2) ( ) = e θ . The irreducible and characteristic series ∀ K − ≡ K L ( ) of the Zeta function is somehow the equivalent of a prime number or an irreducible polynomial, K in Bézout’s studies. This series is at the source of the holomorphic Zeta function. It structures the Ksi function and ensures the localization of non-trivial zeros on the critical line.

4.3. Synthesis The summary of formulas in critical strip is as follows: S

s = x + iy; 0 x 1; y > 0; t = y/2π; ey = t log(t 1); sˆ = 1 s; ≤ ≤ 1 − − s : ζ(s) = ξ(s)ζ(sˆ); ξ(sˆ) = ξ− (s) ∀ 2 θ1 = π/4; θ2 = 2π ey ; θ3 = 1/24y; θ4 = (x 1/2) /2y − − 1/2 x − − α = θ1 + θ2 = 2π(1/8 ey ); k = t − ; ω = α + β = α + θ s 1/2 − 1/2 x i(θ +θ ) i(θ +θ ) iα S iω ξ(s) = π − Γ(sˆ/2)/Γ(s/2) = t − e 1 2 e˘ 3 4 = ke (s) = ke 1/2 x K k = t − ; ω = α + β = α + θ S eˇi(θ3+θ4) = (s) = eiθ3+ iθ 4 + 𝓀 (s) K β = arg( (s)) = θ (x, y) = θ3 + θ4 + θ(x, y) K S 3 (s) = (x 1/2)κ(s)/y + iδ(y)/y − ( ) = P ( )k P2k+1 m κ s i k∞=0 i/y m=0 amx 2 2 3 4 5 2 = i(x 1/2) + 1 + 10x 8x /48y + i 5/12 x + 23x /6 x + x 2x /48y − − − − − − − + 133/720 + 2753x/360 + 707x2/60 181x3/10 87x4/10 5x5 + 12x6 /242 y3+ ℴ 1/y4 P − − − δ(y) = ikb /yk = 7/ 5 242 1 i/y + 1/y2 k∞ =0 k − × iθ − 3 (s)modulo(x 1/2) = e 3 + iδ(y)/y K −

The summary of the formulas on the critical line is as follows: L ξ( ) = π yΓ ˆ /2 /Γ( /2) = ei(θ1+θ2)eˇiθ3 = eiα (s) − K eˇiθ3 = ( ) = eiθ3 + (y) = eiθ K L β = arg( ( )) = θ = θ3 + θ(y) K L (y) = θ 3/5 7 24i 7 24θ + (17807 12/7)iθ 2 (4 2487259/7 45)θ 3 3 × − × 3 × 3 − × × 3 +(11 41 75654497/8 7 45)iθ 4 × × × × 3 +(47 108544391/7 45)θ 5 + 1/y9 × × 3

The decomposition architecture of angles in the critical strip and on the critical line, with the associated series angles, is summarized in Table3.

Table 3. The successive rotations.

Rotation Angle Digressive Angle Critical Strip Critical Line pure π/4 0 θ θ 1 1 pure 2π ey 0 θ2 θ2 − digressive 1/24y θ˘3 θ = θ = θ3 + θ(y) − 2 S L digressive (x 1/2) /2y θ˘4 θ3 + θ4 + θ(x, y) N.A. − −

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Our method extracts and disconnects from the Ksi function ξ(s) = k eiα (s), a homothety 1 × × K 2 y /2 x P π 1 (x 1/2) of ratio k = − and pure rotations of angle θj = 2π ey − , since the 2π 4 − − 24y − 2y subgroup of the group of similarities, composed of homotheties and rotations, leaves non-trivial zeros invariant: ζ(s) = 0 ζ(s) k 1e iθ = 0. The analysis is concentrated on the remainder ξ(s) (s), ⇒ × − − ≡ K in particular the existence of a potential symmetry in x and 1 x, in the series of the remainder (s). − K The Ksi function congruence, via this homothety and these rotations, is justiﬁed by a fundamental property of the Zeta function, which is the existence of an equivalence relation on the y axis. We split y 1 n n 2+ the critical line : x = /2 into contiguous intervals In = [yn, yn+1); n N; 2π = −n 2+ . In the L ∈ log −e interval In, we ﬁnd the nth non-trivial zero with high probability. This segmentation organizes an equivalence relationship taking into account the fractional part ey = ey [ey] of the anamorphosed y y − π variable ey = log , which we associate with two rotations 2( 2π ey ) 1 of angle 2π 2πe R − ◦ R 4 π n y y o α = θ1 + θ2 = 2π log = 2π(1/8 ey ). These rotations drop the Ksi function undulations. 4 − 2π 2πe − This rotation angle 2 1 corresponds, on the critical line, to the alternative undulations (real part R ◦ R (ζ( )) and imaginary part) of the function ξ(𝓈) = eiαeˇiβ eiα. The ratio % = = tan α also structures (ζ( )) 2 < the Zeta function, with the particular values of ey from Table4:

Table 4. Remarkable (ζ( )) versus (ζ( )) relationship. = < ~ {y} 1/8 3/8 5/8 7/8 (ζ( )) 0 (ζ( )) (ζ( )) 0 (ζ( )) %= / = = < = = < 1/% 48y (ζ( )) % 48y (ζ( )) −< <

This angle α arises in the distribution function ey of the presence of non-trivial zeros and in the ratio of the Gamma functions, at the origin of the Ksi function. The ratio 48y is derived, among other things, from the Bernoulli number 2 = 1/6 in the Gamma function series expansion. The rotation 1 π B R of angle θ1 = 4 dissolves the ﬁrst term bias in the Riemann series. The demonstration outline is therefore as follows. We are looking for a class of functions ϕ, which cancels out the Zeta function, in order to identify the kernel of the morphism ϕ ζ. If such ◦ a function exists, it must satisfy the identity ϕ(ζ(s)) = 0 for the pair of points M and Mˆ with coordinates s := (x, y) and sˆ = 1 s := (1 x, y). Unfortunately the way in which the x and y coordinates are − − − entangled at the heart of the Zeta function and of the Ksi function make it impossible to disentangle them, in order to exhibit a function ϕ in the whole critical strip, both for the pair (x, y) and for the pair (1 x, y). Both non-separable coordinates in the ratio Γ sˆ /Γ s unfortunately make this operation − − 2 2 fruitless. This inability to decompose into unbreakable components of the relationship is evident in the expansion into series (s). It is also explained by examining the Zeta derivative whose term K n sLog(n) reveals that each term in the series muddles up coordinates with regard to the term n s. − − − This impossibility is also interpreted by the holomorphic function structure which is a conformal transformation (by locally preserving angles) whose angles in the critical strip are intimately linked to the unbreakable pair (x, y). On the other hand, when x = 1/2, the ratio becomes Γ(s/2)/Γ(s/2) and i(α+θ ) the angle β simply depends on y, so that ζ( ) = e L ζ( ). So, there is indeed a relation on the critical line au + bv = 0: y y 1 i(α+θ ) 1 = /2 + iy, ϕ : ϕ ζ = ζ ζ e L = 0; α = 2π( /8 ey ); ey = log ∀ ∃ ◦ − − 2π 2πe

The angle series θ , a function of y, of sum function θ3, is written as: L θ = θ3 + θ(y); warning: arg( ( )) = θ θ3 + arg 𝓀 (y) 1/24y L K L ≈ −

= 1/2 + iy, (u, v) : (ζ( )) u (y) + (ζ( )) v(y) = 0. ∀ ∃ < × = ×

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In the class of functions ϕ, the two series u = 1 cos(α + θ ) ; v = sin(α + θ ), or u = − L − L sin(α + θ ); v = 1 + cos(α + θ ), may be chosen, since in an underlying way, the relation − L L (ζ(𝓈)) sin((α + θ )/2) = (ζ( )) cos((α + θ )/2) governs the stability of both elements. The RH < L = L seems to be strengthened, since this identity is null only for x = 1/2. The formula, decomposed into Ksi series, splits the inﬂuence of x and y coordinates: On the one 1/2 x hand, the amplitude of ξ(s) is k = t − , and on the other hand, the phase (the argument) of ξ(s) is more diﬃcult to analyze because the pair (x, y) is indissoluble. The deforming mechanism between M and Mˆ is generated by a similarity whose rotation angle ω = α + β is generated by the two pure rotations 2 1 of angle α and two digressive rotations 4 3 of angle β = θ (x, y) = θ3 + θ4 + θ(x, y). R ◦ R R ◦ R S The angle α = π/4 2π ey of rotations 2( 2π ey ) 1(π/4) depends only on the fractional part ey − R − ◦ R 2 1 (x 1/2) of the anamorphosis of y, and the angle β = θ = 24y −2y + θ(x, y) depends inextricably on the S − − pair (x, y). By the deformation of these lenses, the functions Γ(s/2) and Γ(sˆ/2) become out of sync, because of the angle β, more precisely because of arg 𝓀 (s) , and this crisscross does not allow the conception of a Bézout identity between both parts, real and imaginary, of the Zeta function. However, 1 1 the entanglement (x, y) and the complexity of the angle β = θ = 24y + θ(y) 24y disappear for L − − 1 1 i(α+θ ) iα iθ3 x = /2 : ξ() = ξ( /2+iy) = e L = e e˘ . The angle β no longer depends on x at all. In the critical strip, the functional equation makes it possible to write the Zeta function, within a homothety and a rotation, in a congruent manner:

1/2 x iα iθ i(θ +θ ) ζ(s) = ξ(s)ζ(sˆ) = t − e e ζ(sˆ) = ζ(s) e˘ 3 4 ζ(sˆ) S ⇒ ≡ The ratio of the two equivalent Zeta functions is a digressive series, which is unfortunately not 2 a composition of perfect rotations = exp( i/24y) exp i(x 1/2) /2y . The remainder is R4 ◦ R3 − × − − a digressive series (s), the ﬁrst order of which shows that the sum function of the angle inevitably entangles x and y. This residue (s) prevents the conception of a relation au + bv = 0 in the critical strip (except ), but gives its holomorphic property to the Zeta function (conformal transformation: the S L angles are locally preserved, the small circle image is a circle). The characteristics (Riemann functional equation and Hadamard product formula) make it possible to distinctly separate the properties on both complementary sets, and . On the other hand, on the critical line, this remainder becomes L S\L (y), which is absorbed in a multiplicative digressive series, since ζ( ) e˘iθ3 ζ( ), and this property ≡ makes it possible to write a relation au + bv = 0.

5. Discussion With this computational approach, families of mathematical beings appear larger than those one usually perceived through mathematical formulas. For this reason, it is necessary to introduce new concepts such as digressive series to take into account the generality of series which appear thanks to calculation. This presentation is stripped of the theoretical envelope that generally surrounds advances on the Zeta function. It emphasizes the numerical point of view, focusing on the properties underlying this function. However, the theoretical insights described below can still be discerned in an underlying manner. The RH poses a question about ﬁnding the complex function roots, by the equation ζ(s) = 0. Finding the polynomial roots involves analyzing the symmetric functions of its roots and breaking the symmetry, if possible. This is how E. Galois showed the impossibility of calculating, in the general case, the roots of a polynomial of degree 5. This is also how Hadamard established the Zeta function ≥ product formula. When investigating the methods carried out to ﬁnd the Riemann function roots in the bibliography, some strategies for proving the RH arise.

5.1. Analytical Perspective: Direct Localization of Zeros in the Critical Strip In historical and recent publications [14–20], the most used strategy is the direct search for non-trivial zeros using the properties of complex analysis or number theory. It is a question of identifying domains of the C space where the zeros could lie in probability, according to the estimate

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of the value of ζ(s) or ζ(s) in these domains or according to the probable locations of zero crossings of both real and imaginary curves. The classic tools (Fourier transform, complex analysis, Dirichlet series) are used. This method leads to approximations and inequalities that make it possible to delimit increasingly narrow domains, and the increasingly strong probabilities of the presence of zeros on the critical line. It is clear that the C space is already an abstraction above numerical calculations: The C space already causes a detour which departs from the reality of arithmetic numbers. Thus, computations developed using holomorphic function properties give a more distant reflection of the mathematicalInformation 2020, 11 authenticity, x FOR PEERof REVIEW “real” functions (those that are calculated), which complicates the29 zeros’ of 32 identification, which are irreparably localized irregularly on the critical line, and hypothetically around thisidentification, line. which are irreparably localized irregularly on the critical line, and hypothetically around this line. 5.2. Algebraic Perspective: Kernel of a Family of Holomorphic Function Series 5.2. Algebraic Perspective: Kernel of a Family of Holomorphic Function Series In this article, a set-theoretical approach has been preferred. This approach takes a step back from the ZetaIn this function article, and a immersesset-theoretical it in aapproach larger function has been space. preferred. It is an algebraicThis approach approach takes to identifya step back the kernel,from the a subsetZeta function of the elements and immerses that are it projected in a larger on thefunction neutral space. arrival It element.is an algebraic We are approach lookingfor to aidentify morphism the kernel,ϕ whose a kernelsubset isof the the whole elements critical that line. are Theprojected image on by the the neutral morphism arrivalϕ of element. the critical We strip are (exceptlooking the for criticala morphism line) is di whosefferent kernel from 0. is the whole critical line. The image by the morphism of the critical strip (except the critical line) is different from 0. x = 1/2; 𝓈 = x + i y : ϕ ζ( ) = 0; x , 1/2; s = x + i y : ϕ ζ(s) , 0 = ½; = + ∶ ◦ ∘ () =0; ≠ ½; = + ∶ ◦ ∘ () ≠0

The kernel is then interpreted as an equivalence relation. The kernel of the compound morphism The kernel is then interpreted as an equivalence relation. The kernel of the compound morphism ϕ ζ is Ker(ϕ ζ) = ζ 1(Ker(ϕ)). Furthermore, the zeros on this critical line can be calculated. If the ∘◦ is (∘◦ ) = −(). Furthermore, the zeros on this critical line can be calculated. If transformation ϕ is linear, we can think of a succession of rotations that try to cancel the linear the transformation φ is linear, we can think of a succession of rotations that try to cancel the linear combination of the real part and the imaginary part of the image. In our case (Figure 15), we use four combination of the real part and the imaginary part of the image. In our2 case (Figure 15), we use four iπ/4 i( 2π ey ) i( 1/24y) i(x 1/2) /2y rotations 1 = e /, 2 = e − ({ } {,})3 = e − (⁄)and 4 = e− − (½). By⁄ this computer process, rotations Rℛ = R, ℛ = R , ℛ = R and ℛ = . By this computer we already obtain a potential zeros’ domain, which is not the critical line, but which includes it and process, we already obtain a potential zeros’ domain, which is not the critical line, but which includes which forms a strip that becomes thinner and thinner as y grows. This domain is already much more it and which forms a strip that becomes thinner and thinner as grows. This domain is already much restricted than those previously obtained by classical methods. more restricted than those previously obtained by classical methods.

Figure 15. TheThe four four rotations: rotations: two two pure pure rotations rotations and and tw twoo digressive digressive rotations: rotations: angle angle scale scale is not is notrespected. respected.

This strategy formally consists of the fact of broadeningbroadening the root search formulation and of more generally studying the the divisibility divisibility of of numerical numerical series series.. Then Then a Zeta a Zeta function function congruence congruence is defined, is defined, to towithin within a homothety a homothety and and multiple multiple rotations. rotations. These These transformations transformations leave leave invariant invariant the location the location of non- of non-trivialtrivial zeros. zeros. The powerful The powerful arsenal arsenal of divisibility of divisibility of arithmetic of arithmetic numbers numbers and polynomials and polynomials can be used, can becombined used, combined with theories with theoriesstemming stemming from the from works the worksof Étienne of Étienne Bézout, Bé zout,in order in order to discover to discover the theproperties properties of ideals of ideals and andrings rings which which allow allow us to us define to define equivalence equivalence classes classes and quotient and quotient spaces. spaces. This Thisapproach approach focuses focuses directly directly on onseri series’es’ properties properties without without first first tackling tackling the the analytical analytical questions of findingfinding zeros.zeros. In practice, we start by analyzing moremore closelyclosely somesome seriesseries ofof trigonometrictrigonometric functions.functions. X X X ∞ ∞ ∞ a(()t) un (t()+) +()b(t) vn (t())==()c(t) w n(()t) n=1 n=1 n=1

In our study, equivalence classes have been defined on the terms and in order to find remarkable properties on the limits of the series 〈〉 and 〈〉. In this way, the idea of numerical series’ divisibility emerges, as well as the morphism kernel concept on the Zeta function. Cauchy defined the product of two series, and therefore, the division of two series can be defined when this operation is possible and when it leads to two converging series.

6. Conclusions It is conceptually difficult to mathematically conceive of a set of points on a complex surface, isolated from each other in a domain of the ℂ plane, especially if these regular points have no characteristics other than being located on the 0-level contours of the intersection of two wavy surfaces, real and imaginary, locally smooth and without singularity. One solution is to wrap this set

Information 2020, 11, 237 29 of 31

In our study, equivalence classes have been defined on the terms un and vn in order to find remarkable properties on the limits of the series un and vn . In this way, the idea of numerical series’ h i h i divisibility emerges, as well as the morphism kernel concept on the Zeta function. Cauchy defined the product of two series, and therefore, the division of two series can be defined when this operation is possible and when it leads to two converging series.

6. Conclusions It is conceptually difficult to mathematically conceive of a set of points on a complex surface, isolated from each other in a domain of the C plane, especially if these regular points have no characteristics other than being located on the 0-level contours of the intersection of two wavy surfaces, real and imaginary, locally smooth and without singularity. One solution is to wrap this set of isolated points into a connected superset and to think about the algebraic properties of this superset. In the case of the Riemann function, the critical line is the superset that covers non-trivial zeros. The existence of a class of functions ϕ ζ in the critical strip of which the kernel Ker(ϕ ζ) is the critical line makes it ◦ ◦ possible to complete the proof, without attempting to determine these zeros more precisely. Following this step, by using the analytical properties of the points of this function, non-trivial zeros can be identified and calculated. In conclusion, the article presents a RH analysis with numerical computation methods. The study contribution follows from the examination of the two-angle conjunction of the Zeta function and of the Ksi function. On the one hand, the Zeta function has a remarkable property on the critical line: The real/imaginary ratio of the Zeta function is interpreted as the angle tangent that generates an equivalence class along the y axis. On the other hand, the Ksi function is congruent, to within a homothety and a rotation, to a remainder series, whose sum function angle is not symmetrical with respect to the line x = 1/2. On the contrary, the residual congruent series intrinsically contains this symmetry on the line x = 1/2. This trigonometric property of the remainder series undoubtedly exists only on the critical line. It is therefore essentially an angular functional constraint of the holomorphic Zeta function and of the associated Ksi function, which is at the origin of the property of the RH. sˆ s This constraint enigma is contained in the Gamma functions’ ratio Γ 2 /Γ 2 . The calculations made it possible to highlight, by a ring morphism ϕ ζ = 0, the formula i(α+θ ) ◦ ζ = ζ e L , valid only on the critical line, which brings out an irreducible numerical series. This formula is an identity au + bv = 0 representing a synchronous balance between both parts (a, b), real and imaginary, of the Zeta function. This equipoise (k = 1; ξ(s) e˘iθ3 = eiβ; β = θ ) is stable ≡ L along the critical line ϕ ζ = ζ ζ eiαe˘iθ3 = 0 , and topples outside of it, in the critical strip. ◦ − On the critical line, the elements a and b of the Zeta function cancel each other out alternately and 2π(n 2+5/8) periodically, a in the middle of the interval I ; n ; y = − ; ζ(1/2+iy ) 0 + ib, and b at n N n log(n 2+5/8) 1 n ∈ − − 2π(n 2+1/8) the start of the interval y = − ; ζ(1/2+iy ) a + i0. n log(n 2+1/8) 1 n Stability allows both parts to systematically− − cancel themselves out at the same time, at an additional point, a non-trivial zero, in the neighborhood of each periodic and alternative zeroing of both parts (a, b). This cyclical phenomenon thus attests to the existence of equivalence classes of the zeroing of both surfaces according to the angle α + β = α + θ . It also substantiates the occurrence number of L these representatives of equivalence classes, that is to say the potential number of zeros on this line ℵ0 . L In the critical strip, the Zeta function happens as two surfaces intersect irregularly. A yi anamorphosis makes it possible to restore quasi-periodic waves. These waves are reflected isometrically + on both sides x = 1/2 and x = 1/2− of the critical line, which act as a rectilinear mirror (Figure 16). On the contrary, these waves are distorted by the prism of a convex lens that emphasizes the folds for x < 1/2, and a concave lens that diminishes them for x > 1/2. The Riemann functional equation provides the distortion link between the two lenses. The congruent transformation in the subgroup of similarities of the Ksi function makes it possible to remove the structural waves and reveal its background, which is a power series 𝓀 (s) whose angle is not symmetrical in x and 1 x. −

5 Information 2020, 11, x FOR PEER REVIEW 30 of 32 of isolated points into a connected superset and to think about the algebraic properties of this superset. In the case of the Riemann function, the critical line is the superset that covers non-trivial zeros. The existence of a class of functions ∘ in the critical strip of which the kernel (∘) is the critical line makes it possible to complete the proof, without attempting to determine these zeros more precisely. Following this step, by using the analytical properties of the points of this function, non-trivial zeros can be identified and calculated. In conclusion, the article presents a RH analysis with numerical computation methods. The study contribution follows from the examination of the two-angle conjunction of the Zeta function and of the Ksi function. On the one hand, the Zeta function has a remarkable property on the critical line: The real/imaginary ratio of the Zeta function is interpreted as the angle tangent that generates an equivalence class along the axis. On the other hand, the Ksi function is congruent, to within a homothety and a rotation, to a remainder series, whose sum function angle is not symmetrical with respect to the line =½. On the contrary, the residual congruent series intrinsically contains this symmetry on the line =½. This trigonometric property of the remainder series undoubtedly exists only on the critical line. It is therefore essentially an angular functional constraint of the holomorphic Zeta function and of the associated Ksi function, which is at the origin of the property of the RH. This ̂ constraint enigma is contained in the Gamma functions’ ratio Γ Γ . The calculations made it possible to highlight, by a ring morphism ∘=0, the formula = ̅ (ℒ) , valid only on the critical line, which brings out an irreducible numerical series. This formula is an identity + = 0 representing a synchronous balance between both parts (, ), real and imaginary, of the Zeta function. This equipoise (=1;() ≡̆ = ; =ℒ) is stable along the critical line ( ∘ = − ̅ ̆ =0), and topples outside of it, in the critical strip. On the critical line, the elements and of the Zeta function cancel each other out alternately (⅝) and periodically, in the middle of the interval ; ∈ℕ; = ;(½+ ) ≅0+, (⅝) (⅛) and at the start of the interval = ;(½+ ) ≅+0. (⅛) Stability allows both parts to systematically cancel themselves out at the same time, at an additional point, a non-trivial zero, in the neighborhood of each periodic and alternative zeroing of both parts (, ). This cyclical phenomenon thus attests to the existence of equivalence classes of the zeroing of both surfaces according to the angle +=+ℒ. It also substantiates the occurrence number of these representatives of equivalence classes, that is to say the potential number ℵ of zeros on this line ℒ. In the critical strip, the Zeta function happens as two surfaces intersect irregularly. A anamorphosis makes it possible to restore quasi-periodic waves. These waves are reflected isometrically on both sides =½⁺ and =½⁻ of the critical line, which act as a rectilinear mirror (Figure 16). On the contrary, these waves are distorted by the prism of a convex lens that emphasizes the folds for <½, and a concave lens that diminishes them for >½. The Riemann functional equation provides the distortion link between the two lenses. The congruent transformation in the Informationsubgroup2020 of, similarities11, 237 of the Ksi function makes it possible to remove the structural waves30 ofand 31 reveal its background, which is a power series () whose angle is not symmetrical in and 1−.

FigureFigure 16.16. The lenses of thethe functionalfunctional equation.equation.

From Sainte-Pélagie prison, in 1831, Évariste Galois [33], pioneer of the set theory outlook, mocking those who got lost in the maze of quadratures, advised researchers to “not jump with both feet into the calculations”. May this article prove that we have learned the lesson against this usual shortcoming, and that we have, at each step of the process, geometrically interpreted the transformations and clariﬁed the path taken by the method.

Funding: This four-year research received no external funding. Acknowledgments: C Fagard-Jenkin, PhD student of mathematics at St Andrews University, acted as an initial proofreader before submission. Conﬂicts of Interest: The author declares no conﬂict of interest.

References

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