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

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The Two-Layer Hierarchical Distribution Model of Zeros of Riemann’S Zeta Function Along the Critical Line information Article The Two-Layer Hierarchical Distribution Model of Zeros of Riemann’s Zeta Function along the Critical Line Michel Riguidel Department of Computer Science and Networks, Télécom ParisTech, 75015 Paris, France; [email protected]; Tel.: +33-67687-5199 Abstract: This article numerically analyzes the distribution of the zeros of Riemann’s zeta function along the critical line (CL). The zeros are distributed according to a hierarchical two-layered model, one deterministic, the other stochastic. Following a complex plane anamorphosis involving the Lambert function, the distribution of zeros along the transformed CL follows the realization of a stochastic process of regularly spaced independent Gaussian random variables, each linked to a zero. The value of the standard deviation allows the possible overlapping of adjacent realizations of the random variables, over a narrow confidence interval. The hierarchical model splits the z function into sequential equivalence classes, with the range of probability densities of realizations coinciding with the spectrum of behavioral styles of the classes. The model aims to express, on the CL, the coordinates of the alternating cancellations of the real and imaginary parts of the z function, to dissect the formula for the number of zeros below a threshold, to estimate the statistical laws of two consecutive zeros, of function maxima and moments. This also helps explain the absence of multiple roots. Keywords: Riemann’s zeta function; non-trivial zeros; Lambert’s function; Laplace–Gauss law; gaps; multiple roots Citation: Riguidel, M. The Two-Layer Hierarchical Distribution 1. Introduction Model of Zeros of Riemann’s Zeta This article presents a distribution model of the non-trivial zeros of the zeta function Function along the Critical Line. on the critical line (CL). The Riemann Hypothesis (RH) [1] places them exclusively on this Information 2021 12 , , 22. https:// line, but this article focuses on the CL zeros, regardless of the RH. doi.org/10.3390/info12010022 We first explored the global behavior of the z function on the CL, and the role of the Lambert function [2] on the folds of the z function in the critical strip (CS). A close Received: 11 December 2020 relationship exists, on the CL, between Riemann’s z function and Lambert’s W function, Accepted: 4 January 2021 Published: 8 January 2021 after an anamorphosis of the ordinate of the complex plane. This article also shows that, on the CL, there is a relationship between both the real Publisher’s Note: MDPI stays neutral and imaginary components of the z function. This ratio, only valid on the CL, takes into with regard to jurisdictional claims in account an angle tied to the fractional part of the anamorphosed ordinate, which makes it published maps and institutional affil- possible to anchor the common zeros of these two components in an equivalence relation, iations. where the classes are intervals of unit length. Thus, both z surfaces, alternatingly zero, with period one on the anamorphosed ordinate of the complex plane, locally generate an additional common zero which is the nth non-trivial zero. The article then investigates, on the anamorphosed axis, the local “random” character of the nth non-trivial zero, in the form of the realization of a Gaussian random variable (RV) Copyright: © 2021 by the author. n s W( n) Licensee MDPI, Basel, Switzerland. with a mean and standard deviation en increasing very slowly with ln , of order 1 3 < 7 This article is an open access article / for n 10 , all these n RVs being pairwise independent. The magnitude of the distributed under the terms and standard deviation leads to the possible permutation of some realizations of adjacent RVs, conditions of the Creative Commons justifying the uncertainty in the formula for the number of zeros below a given threshold, Attribution (CC BY) license (https:// and explaining the corresponding difficulty for algorithms to compute these zeros. creativecommons.org/licenses/by/ Finally, this article presents a taxonomy of the local behaviors of the z function in the 4.0/). CS, according to the probability density of the RV relevant to the nth zero. The model still Information 2021, 12, 22. https://doi.org/10.3390/info12010022 https://www.mdpi.com/journal/information Information 2021, 12, x FOR PEER REVIEW 2 of 35 Information 2021, 12, 22 Finally, this article presents a taxonomy of the local behaviors of the function in 2 of 35 the CS, according to the probability density of the RV relevant to the th zero. The model still allows a number of derivations which reaffirm some classical formulas of the Rie- mann function. This also helps explain the absence of multiple roots. allows a number of derivations which reaffirm some classical formulas of the Riemann z By consulting the abundant scientific literature on the zeros of the Riemann func- function. This also helps explain the absence of multiple roots. tion, including certain pioneering articles [3–8], one can distinguish three broad families By consulting the abundant scientific literature on the zeros of the Riemann z function, of publications, with the first two having the underlying ambition to provide proof of the including certain pioneering articles [3–8], one can distinguish three broad families of RH. publications, with the first two having the underlying ambition to provide proof of the RH. The first family consists in circumscribing the perimeter of the region where the zeros The first family consists in circumscribing the perimeter of the region where the zeros are located. As early as 1896, Hadamard [3] and La Vallée Poussin [4] restricted the “fertile are located. As early as 1896, Hadamard [3] and La Vallée Poussin [4] restricted the “fertile 0<<1 region” ofregion” zeros to of the zeros CS to the CS (0 <. Then,x < 1 during). Then, the during twentieth the twentieth century, the century, region the was region was reduced inreduced two ways. in two On ways. the one On hand, the one by hand,further by limiting further the limiting perimeter, the perimeter, and on the and other on the other hand by handincreasing by increasing the percentage the percentage of potential of potentialzeros on zerosor around on or the around CL [9,10], the CL in [9an,10 ], in an increasinglyincreasingly restricted restricted perimeter. perimeter. The secondThe family second consists family in consists analyzing in analyzing only CL zeros only CL[11,12], zeros by [11 estimating,12], by estimating the num- the num- ber of zerosber ofless zeros than less a than value a y ofvalue the ofordinate the ordinate of [13], of s [by13 ],calculating by calculating the number the number of of zero zero crossingscrossings on the on theCL, CL,by studying by studying the the adja adjacentcent zeros, zeros, or orby by estimating estimating other other charac- characteristics teristics ofof the z function, function, always always with with the the aim aim of apprehending of apprehending theRH, the accordingRH, according to various to points various pointsof view. of view. The thirdThe family, third more family, concretely, more concretely, consists in consists effectively in effectively calculating calculating the CL zeros the CLwith zeros with a numericala numerical algorithm algorithm [14,15]. Currently, [14,15]. Currently, the exercise, the exercise,which thrived which between thrived between1985 and 1985 and 2005, has2005, seen hasa loss seen of amomentum, loss of momentum, as the nu asmber the number of known of knownzeros is zeros substantial. is substantial. Extend- Extending ing thesethese calculations calculations has become has become superfluous superfluous with withsilicon silicon computers, computers, although although searches searches for for zeros zerosbeyond beyond aboutabout > 10n > 10over100 anover interval an interval of the of order theorder of Δ of =D 10n =, 10would10, would pro- provide vide interestinginteresting additional additional knowledge knowledge on the on the function.z function. The aim ofThe this aim article of this belongs article to belongs the second to the family, second by family, taking by ad takingvantage advantage of numerical of numerical results ofresults the third of thefamily. third Indeed, family. the Indeed, goal here the goal is not here to isinvestigate not to investigate questions questions about the about the RH, nor RH,to determine nor to determine an algorithm an algorithm to effectively to effectively calculate calculatezeros. The zeros. goal Theis mainly goal is to mainly to define, ondefine, the CL, on a the distribution CL, a distribution model with model two with layers, two one layers, deterministic one deterministic and the other and the other stochastic,stochastic, to specify to the specify local thecharacteristics local characteristics of the function of the z functionaround the around zeros the and zeros to and to discuss thediscuss evolution the evolution of local behavior of local behaviorof rare events of rare with events increasing with increasing . n. Despite itsDespite age, Edward its age, Charles Edward Titchmarsh’s Charles Titchmarsh’s book [16] book is the [16 most] is thecomprehensive most comprehensive book on book function on z function theory. theory. It covers It covers almost almost everything everything anyone anyone would would want want to know to know about about thethe zfunction.function. Aleksandar Aleksandar Ivi Ivi´c’sbookć’s book ( (AАлександарлексaндap Ивић))[ [17]17] offersoffers anan easier easier approach approachfor for
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