A study of the Riemann zeta function

The functional equation for Riemann's Zeta function is studied, from which it is shown why all of the non-trivial, full-zeros of the Zeta function $\zeta (s)$ will only occur on the critical line {$\sigma=1/2$} where {$s=\sigma+I \rho$}, thereby establishing the truth of Riemann's hypothesis. Further, two relatively simple transcendental equations are obtained; the numerical solution of these equations locates all of the zeros of {$\zeta (s)$} on the critical line. Comment: 8 pages, 3 figures

arXiv, 2020

In this study, I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeroes of the Riemann Zeta-Function is the critical line given by ℜ(s)=1/2. The methods and results of this paper are based on well-known theorems on the number of zeroes for complex-valued functions (Jensen’s, Titchmarsh’s and Rouché’s theorem), with the Riemann Mapping Theorem acting as a bridge between the Unit Disk on the complex plane and the critical strip. By primarily relying on well-known theorems of complex analysis our approach makes this paper accessible to a relatively wide audience permitting a fast check of its validity. Both proofs do not use any functional equation of the Riemann Zeta-Function, except leveraging its implied symmetry for non-trivial zeroes on the critical strip.

2020

I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeroes of the Riemann Zeta Function is the critical line. Methods and results of this paper are based on well-known theorems on the number of zeroes for complex value functions (Jensen, Titchmarsch, Rouche theorems), with the Riemann Mapping Theorem acting as a bridge between the Unit Disk on the complex plane and the critical strip. By primarily relying on well-known theorems of complex analysis our approach makes this paper accessible to a relatively wide audience permitting a fast check of its validity. Both proofs do not use any functional equation of the Riemann Zeta Function, except leveraging its implied symmetry for non-trivial zeroes on the critical strip.

2024

The Riemann hypothesis, stating that all nontrivial zeros of the Riemann zeta function have real parts equal to 1/2, is one of the most important conjectures in mathematics. In this paper we prove the Riemann hypothesis by adding an extra unbounded term to the traditional definition, extending its validity to Rez>0. The Stolz-Cesàro theorem is then used to analyse zeta(z)/zeta(1-z) as a ratio of complex sequences. The results are analysed in both halves of the critical strip (0<Rez<1/2, 1/2<Rez<1), yielding a contradiction when it is assumed that zeta(z)=0 in either of these halves.

Mathematics and Statistics, 2022

The Riemann zeta (ζ) function ζ(s) = ∞ n=1 1 n s is valid for all complex number s = x + iy : Re(s) > 1, for the line x = 1. Euler-Riemann found that the function equals zero for all negative even integers: −2, −4, −6, • • • (commonly known as trivial zeros) has an infinite number of zeros in the critical strip of complex numbers between the lines x = 0 and x = 1. Moreover, it was well known to him that all non-trivial zeros are exhibiting symmetry with respect to the critical line x = 1 2. As a result, Riemann conjectured that all of the non-trivial zeros are on the critical line, this hypothesis is known as the Riemann hypothesis. The Riemann zeta function plays a momentous part while analyzing the number theory and has applications in applied statistics, probability theory and Physics. The Riemann zeta function is closely related to one of the most challenging unsolved problems in mathematics (the Riemann hypothesis) which has been classified as the 8th of Hilbert's 23 problems. This function is useful in number theory for investigating the anomalous behavior of prime numbers. If this theory is proven to be correct, it means we will be able to know the sequential order of the prime numbers. Numerous approaches have been applied towards the solution of this problem, which includes both numerical and geometrical approaches, also the Taylor series of the Riemann zeta function, and the asymptotic properties of its coefficients. Despite the fact that there are around 10 13 , non-trivial zeros on the critical line, we cannot assume that the Riemann Hypothesis (RH) is necessarily true unless a lucid proof is provided. Indeed, there are differing viewpoints not only on the Riemann Hypothesis's reliability, but also on certain basic conclusions see for example [16] in which the author justifies the location of non-trivial zero subject to the simultaneous occurrence of ζ(s) = ζ(1 − s) = 0, and omitting the impact of an indeterminate form ∞.0, that appears in Riemann's approach. In this study we also consider the simultaneous occurrence ζ(s) = ζ(1 − s) = 0 but we adopt an element-wise approach of the Taylor series by expanding n −x for all n = 1, 2, 3, • • • at the real parts of the non-trivial zeta zeros lying in the critical strip for s = α + iy is a non-trivial zero of ζ(s), we first expand each term n −x at α then at 1 − α. Then In this sequel, we evoke the simultaneous occurrence of the non-trivial zeta function zeros ζ(s) = ζ(1 − s) = 0, on the critical strip by the means of different representations of Zeta function. Consequently, proves that Riemann Hypothesis is likely to be true.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. [The paper is published in a journal of number theory.]

In this work we consider a new functional equation for the Riemann ζ-function in the critical half-strip S + ≡ s = x + iy ∈ C : 1 2 < x < 1, y > 0. With the help of this equation we prove that finding non-trivial zeros of the Rie-mann ζ-function outside the critical line Re(s) = 1 2 would be equivalent to the existence of complex numbers s = x + iy ∈ S + for which ∞ k=1 (−1) k+1 1 ζ(2x) ∞ n=1 cos y log 1 + k n n 2x 1 + k n x = 1 2. Such a condition is studied, and the attempt of proving the Riemann hypothesis is found to involve also the functional equation χ(t) = −χ t + 1 n , where t is a real variable ≥ 1, and n is any natural number. The limiting behaviour of the solutions χn(t) as t approaches 1 is then studied in detail.

Proceedings of the American Mathematical Society, 1972

Estimates are given for the number of zeros of Re{ír-"sr(i/2)í(i)} and lTr\{Tr-"2T(sl2)r(s)} with 0<Ims<7-, and fixed Re 5 inside the critical strip.

viXra, 2018

This paper explicates the Riemann hypothesis and proves its validity; it explains why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1. Much exact calculations are presented, instead of approximations, for the sake of accuracy or precision, clarity and rigor. (N.B.: New materials have been added to the paper.

This paper explicates the Riemann hypothesis and proves its validity. [The paper is published in a journal of number theory.]