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13 pages, 269 KiB

Open AccessFeature PaperEditor’s ChoiceArticle

On Equivalents of the Riemann Hypothesis Connected to the Approximation Properties of the Zeta Function

by Antanas Laurinčikas

Axioms 2025, 14(3), 169; https://doi.org/10.3390/axioms14030169 - 26 Feb 2025

Viewed by 799

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function

ζ (s)

(zeros different from

s = - 2 m

m \in N

) lie on the critical line

σ = 1 / 2

. [...] Read more.

The famous Riemann hypothesis (RH) asserts that all non-trivial zeros of the Riemann zeta function

ζ (s)

(zeros different from

s = - 2 m

m \in N

) lie on the critical line

σ = 1 / 2

. In this paper, combining the universality property of

ζ (s)

with probabilistic limit theorems, we prove that the RH is equivalent to the positivity of the density of the set of shifts

ζ (s + i t_{τ})

approximating the function

ζ (s)

. Here,

t_{τ}

denotes the Gram function, which is a continuous extension of the Gram points. Full article

8 pages, 728 KiB

Open AccessArticle

On the Approximation of the Hardy Z-Function via High-Order Sections

by Yochay Jerby

Axioms 2024, 13(9), 577; https://doi.org/10.3390/axioms13090577 - 25 Aug 2024

Viewed by 1832

Abstract

The Z-function is the real function given by

Z (t) = e^{i θ (t)} ζ (\frac{1}{2} + i t)

, where

ζ (s)

is the Riemann zeta function, and

θ (t)

is [...] Read more.

The Z-function is the real function given by

Z (t) = e^{i θ (t)} ζ (\frac{1}{2} + i t)

, where

ζ (s)

is the Riemann zeta function, and

θ (t)

is the Riemann–Siegel theta function. The function, central to the study of the Riemann hypothesis (RH), has traditionally posed significant computational challenges. This research addresses these challenges by exploring new methods for approximating

Z (t)

and its zeros. The sections of

Z (t)

are given by

Z_{N} (t) : = \sum_{k = 1}^{N} \frac{cos (θ (t) - ln (k) t)}{\sqrt{k}}

for any

N \in N

. Classically, these sections approximate the Z-function via the Hardy–Littlewood approximate functional equation (AFE)

Z (t) \approx 2 Z_{\tilde{N} (t)} (t)

for

\tilde{N} (t) = [\sqrt{\frac{t}{2 π}}]

. While historically important, the Hardy–Littlewood AFE does not sufficiently discern the RH and requires further evaluation of the Riemann–Siegel formula. An alternative, less common, is

Z (t) \approx Z_{N (t)} (t)

for

N (t) = [\frac{t}{2}]

, which is Spira’s approximation using higher-order sections. Spira conjectured, based on experimental observations, that this approximation satisfies the RH in the sense that all of its zeros are real. We present a proof of Spira’s conjecture using a new approximate equation with exponentially decaying error, recently developed by us via new techniques of acceleration of series. This establishes that higher-order approximations do not need further Riemann–Siegel type corrections, as in the classical case, enabling new theoretical methods for studying the zeros of zeta beyond numerics. Full article

(This article belongs to the Special Issue Numerical Methods and Approximation Theory)

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30 pages, 8350 KiB

Open AccessArticle

An Analytical and Numerical Detour for the Riemann Hypothesis

by Michel Riguidel

Information 2021, 12(11), 483; https://doi.org/10.3390/info12110483 - 21 Nov 2021

Viewed by 2532

Abstract

From the functional equation

F (s) = F (1 - s)

of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function [...] Read more.

From the functional equation

F (s) = F (1 - s)

of Riemann’s zeta function, this article gives new insight into Hadamard’s product formula. The function

S_{1} (s) = d (\ln F (s)) / d s

and its family of associated

S_{m}

functions, expressed as a sum of rational fractions, are interpreted as meromorphic functions whose poles are the poles and zeros of the

F

function. This family is a mathematical and numerical tool which makes it possible to estimate the value

F (s)

of the function at a point

s = x + i y = \dot{x} + ½ + i y

in the critical strip

S

from a point

𝓈 = ½ + i y

on the critical line

ℒ

.Generating estimates

S_{m}^{*} (s)

S_{m} (s)

at a given point requires a large number of adjacent zeros, due to the slow convergence of the series. The process allows a numerical approach of the Riemann hypothesis (RH). The method can be extended to other meromorphic functions, in the neighborhood of isolated zeros, inspired by the Weierstraß canonical form. A final and brief comparison is made with the

ζ

and

F

functions over finite fields. Full article

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50 pages, 578 KiB

Open AccessArticle

Combinatorial Models of the Distribution of Prime Numbers

by Vito Barbarani

Mathematics 2021, 9(11), 1224; https://doi.org/10.3390/math9111224 - 27 May 2021

Cited by 4 | Viewed by 6060

Abstract

This work is divided into two parts. In the first one, the combinatorics of a new class of randomly generated objects, exhibiting the same properties as the distribution of prime numbers, is solved and the probability distribution of the combinatorial counterpart of the [...] Read more.

π (x)

. A proposition equivalent to the Prime Number Theorem (PNT) is proved to hold, while the equivalent of the Riemann Hypothesis (RH) is proved to be false with probability 1 (w.p. 1) for this model. Many identities involving Stirling numbers of the second kind and harmonic numbers are found, some of which appear to be new. The second part is dedicated to generalizing the model to investigate the conditions enabling both PNT and RH. A model representing a general class of random integer sequences is found, for which RH holds w.p. 1. The prediction of the number of consecutive prime pairs as a function of the gap d, is derived from this class of models and the results are in agreement with empirical data for large gaps. A heuristic version of the model, directly related to the sequence of primes, is discussed, and new integral lower and upper bounds of

π (x)

are found. Full article

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31 pages, 9476 KiB

Open AccessArticle

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

by Michel Riguidel

Information 2020, 11(5), 237; https://doi.org/10.3390/info11050237 - 26 Apr 2020

Cited by 1 | Viewed by 3064

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 [...] Read more.

ζ (x + i y) = a + i b

and

ξ (x + i y) = p + i q

, in the critical strip. On the one hand, the two-dimensional surface angle

\tan^{- 1} (b / a)

of the Riemann Zeta function

ζ

is related to the semi-angle of the fractional part of

\frac{y}{2 π} \ln (\frac{y}{2 π e})

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

a u + b v \equiv 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. Full article

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29 pages, 15004 KiB

Open AccessArticle

Morphogenesis of the Zeta Function in the Critical Strip by Computational Approach

by Michel Riguidel

Mathematics 2018, 6(12), 285; https://doi.org/10.3390/math6120285 - 26 Nov 2018

Cited by 3 | Viewed by 6261

Abstract

This article proposes a morphogenesis interpretation of the zeta function by computational approach by relying on numerical approximation formulae between the terms and the partial sums of the series, divergent in the critical strip. The goal is to exhibit structuring properties of the partial sums of the raw series by highlighting their morphogenesis, thanks to the elementary functions constituting the terms of the real and imaginary parts of the series, namely the logarithmic, cosine, sine, and power functions. Two essential indices of these sums appear: the index of no return of the vagrancy and the index of smothering of the function before the resumption of amplification of its divergence when the index tends towards infinity. The method consists of calculating, displaying graphically in 2D and 3D, and correlating, according to the index, the angles, the terms and the partial sums, in three nested domains: the critical strip, the critical line, and the set of non-trivial zeros on this line. Characteristics and approximation formulae are thus identified for the three domains. These formulae make it possible to grasp the morphogenetic foundations of the Riemann hypothesis (RH) and sketch the architecture of a more formal proof. Full article

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