Rigorous Proof for Riemann Hypothesis Using Sigma-Power Laws

Mathematics and Statistics, 2021

In 1859, Bernhard Riemann, a German mathematician, published a paper to the Berlin Academy that would change mathematics forever. The mystery of prime numbers was the focus. At the core of the presentation was indeed a concept that had not yet been proven by Riemann, one that to this day baffles mathematicians. The way we do business could have been changed if the Riemann hypothesis holds true, which is because prime numbers are the key element for banking and e-commerce security. It will also have a significant influence, impacting quantum mechanics, chaos theory, and the future of computation, on the cutting edge of science. In this article, we look at some well-known results of Riemann Zeta function in a different light. We explore the proofs of Zeta integral Representation, Analytic continuity and the first functional equation. Initially, we observe omitting a logical undefined term in the integral representation of Zeta function by the means of Gamma function. For that we propound some modifications in order to reasonably justify the location of the non-trivial zeros on the critical line: = 1 2 by assuming that () and (1 −) simultaneously equal zero. Consequently, we conditionally prove Riemann Hypothesis. MSC 2010 Classification: 97I80, 11M41

In 1859, Georg Friedrich Bernhard Riemann had announced the following conjecture, called Riemann Hypothesis : The nontrivial roots (zeros) s=\sigma+it of the zeta function, defined by: \zeta(s) = \sum_{n=1}^{+\infty}\frac{1}{n^s}, for \Re(s)>1 have real part} \sigma= 1/2. In this note, I give the proof that \sigma=1/2 using an equivalent statement of the Riemann Hypothesis concerning the Dirichlet \eta function.

In 1859, Georg Friedrich Bernhard Riemann had announced the following conjecture, called Riemann Hypothesis : The nontrivial roots (zeros) s=\sigma+it of the zeta function, defined by: \zeta(s) = \sum_{n=1}^{+\infty}1/n^s, \Re(s)>1 have real part \sigma= 1/2. In this note, I give the proof that \sigma= 1/2 using an equivalent statement of the Riemann Hypothesis concerning the Dirichlet eta function.

Problems of the Millennium: the Riemann Hypothesis, 2009

I. The problem. The Riemann zeta function is the function of the complex variable s, defined in the half-plane 1 (s) > 1 by the absolutely convergent series ζ(s) := ∞ n=1 1 n s , and in the whole complex plane C by analytic continuation. As shown by Riemann, ζ(s) extends to C as a meromorphic function with only a simple pole at s = 1, with residue 1 , and satisfies the functional equation π −s/2 Γ(s 2) ζ(s) = π −(1−s)/2 Γ(1 − s 2) ζ(1 − s). (1) In an epoch-making memoir published in 1859, Riemann [Ri] obtained an analytic formula for the number of primes up to a preassigned limit. This formula is expressed in terms of the zeros of the zeta function, namely the solutions ρ ∈ C of the equation ζ(ρ) = 0. In this paper, Riemann introduces the function of the complex variable t defined by ξ(t) = 1 2 s(s − 1) π −s/2 Γ(s 2) ζ(s) with s = 1 2 +it, and shows that ξ(t) is an even entire function of t whose zeros have imaginary part between −i/2 and i/2. He further states, sketching a proof, that in the range between 0 and T the function ξ(t) has about (T /2π) log(T /2π) − T /2π zeros. Riemann then continues: "Man findet nun in der That etwa so viel reelle Wurzeln innerhalb dieser Grenzen, und es ist sehr wahrscheinlich, dass alle Wurzeln reell sind.", which can be translated as "Indeed, one finds between those limits about that many real zeros, and it is very likely that all zeros are real." The statement that all zeros of the function ξ(t) are real is the Riemann hypothesis. The function ζ(s) has zeros at the negative even integers −2, −4,. .. and one refers to them as the trivial zeros. The other zeros are the complex numbers 1 2 + iα where α is a zero of ξ(t). Thus, in terms of the function ζ(s), we can state Riemann hypothesis. The nontrivial zeros of ζ(s) have real part equal to 1 2. In the opinion of many mathematicians the Riemann hypothesis, and its extension to general classes of L-functions, is probably today the most important open problem in pure mathematics. II. History and significance of the Riemann hypothesis. For references pertaining to the early history of zeta functions and the theory of prime numbers, we refer to Landau [La] and Edwards [Ed]. 1 We denote by (s) and (s) the real and imaginary part of the complex variable s. The use of the variable s is already in Dirichlet's famous work of 1837 on primes in arithmetic progression.

Riemann hypothesis is proven, using R. Salem (1953) equivalence and numerical experiments. The breakthrough are new important properties of the Riemann zeta function obtained by numerical experiments.

arXiv: General Mathematics, 2009

Using the $\zeta$ functional equation and the Hadamard product, an analytical expression for the sum of the reciprocal of the $\zeta$ zeros is established. We then demonstrate that on the critical line, $|\zeta|$ is convex, and that in the region $0 < \Re(s) \leq 0.5$, $|\zeta|$ has a negative slope. In each case, analytical formulae are established, and numerical examples are presented to validate these formulae.

Using the mean and mean deviation approach for finding quadratic roots, an energy equation as a homogeneous second order ordinary differential equation combining Newton's and einstein's energy equations involving the speed of light c produces its auxilliary equation's complex roots one of which is used as the required s = 1/2 + it in the Riemann zeta function. The function relies on s which relies on t which relies on mass of particles m and distance k traveled (or frequencies there-of)to generate values. As a series it sums to 0 on t approaching infinity as required.

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.

arXiv: Complex Variables, 2019

In this work we consider a new functional equation for the Riemann zeta-function in the critical half-strip. With the help of this equation we prove that finding non-trivial zeros of the Riemann zeta-function outside the critical line would be equivalent to the existence of complex numbers for which equation (5.1) in the paper holds. Such a condition is studied, and the attempt of proving the Riemann hypothesis is found to involve also the functional equation (6.26), where t is a real variable bigger than or equal to 1 and n is any natural number. The limiting behavior of the solutions as t approaches 1 is then studied in detail.

2021

This paper presents a new approach towards the Riemann Hypothesis. On iterative expansion of integration term in functional equation of the Riemann zeta function we get sum of two series function. At the ‘nontrivial’ zeros of zeta function, value of the series is zero. Thus, Riemann hypothesis is false if that happens for an ‘s’ off the line <(s) = 1/2 ( the critical line). This series has two components f(s) and f(1− s). For the hypothesis to be false one component is additive inverse of the other. From geometric analysis of spiral geometry representing the component series functions f(s) and f(1 − s) on complex plane we find by contradiction that they cannot be each other’s additive inverse for any s, off the critical line. Thus, proving truth of the hypothesis.