Abstract
This article numerically analyzes the distribution of the zeros of Riemann’s zeta function along the critical line (CL). The zeros are distributed according to a hierarchical two-layered model, one deterministic, the other stochastic. Following a complex plane anamorphosis involving the Lambert function, the distribution of zeros along the transformed CL follows the realization of a stochastic process of regularly spaced independent Gaussian random variables, each linked to a zero. The value of the standard deviation allows the possible overlapping of adjacent realizations of the random variables, over a narrow confidence interval. The hierarchical model splits the ζ function into sequential equivalence classes, with the range of probability densities of realizations coinciding with the spectrum of behavioral styles of the classes. The model aims to express, on the CL, the coordinates of the alternating cancellations of the real and imaginary parts of the ζ function, to dissect the formula for the number of zeros below a threshold, to estimate the statistical laws of two consecutive zeros, of function maxima and moments. This also helps explain the absence of multiple roots.
Highlights
This article presents a distribution model of the non-trivial zeros of the zeta function on the critical line (CL)
The Riemann Hypothesis (RH) [1] places them exclusively on this line, but this article focuses on the CL zeros, regardless of the RH
We first explored the global behavior of the ζ function on the CL, and the role of the Lambert function [2] on the folds of the ζ function in the critical strip (CS)
Summary
This article presents a distribution model of the non-trivial zeros of the zeta function on the critical line (CL). This article shows that, on the CL, there is a relationship between both the real and imaginary components of the ζ function This ratio, only valid on the CL, takes into account an angle tied to the fractional part of the anamorphosed ordinate, which makes it possible to anchor the common zeros of these two components in an equivalence relation, where the classes are intervals of unit length. Both ζ surfaces, alternatingly zero, with period one on the anamorphosed ordinate of the complex plane, locally generate an additional common zero which is the nth non-trivial zero. Whetrhee pworsistiinbgle,,soclmasesiocrailgimnaalthneomtaatitoicnasl anroetdaetifionnesdh. ave been selected (Table 1): to lighten the writing, some original notations are defined
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