Prime Gaps, Zeta Function Behavior, and Counterexamples to the Riemann Hypothesis

2010

In this paper, first by employing inequalities derived from the Opial inequality due to David Boyd with best constant, we will establish new unconditional lower bounds for the gaps between the zeros of the Riemann zeta function. Second, on the hypothesis that the moments of the Hardy Z−function and its derivatives are correctly predicted, we establish some explicit formulae for the lower bounds of the gaps between the zeros and use them to establish some new conditional bounds. In particular it is proved that the consecutive nontrivial zeros often differ by at least 6.1392 (conditionally) times the average spacing. This value improves the value 4.71474396 that has been derived in the literature.

In this study, the Riemann problem is presented with highlights on history of the zeta function.

Eprint Arxiv 1001 0494, 2010

In this paper, first by employing inequalities derived from the Opial inequality due to David Boyd with best constant, we will establish new unconditional lower bounds for the gaps between the zeros of the Riemann zeta function. Second, on the hypothesis that the moments of the Hardy Z-function and its derivatives are correctly predicted, we establish some explicit formulae for the lower bounds of the gaps between the zeros and use them to establish some new conditional bounds. In particular it is proved that the consecutive nontrivial zeros often differ by at least 6.1392 (conditionally) times the average spacing. This value improves the value 4.71474396 that has been derived in the literature.

2021

The Riemann zeta function is one of the most important functions in mathematics. In this paper I will demonstrate the relation of the Riemann zeta function to the prime numbers. In the latter we will use this function to prove the prime number theorem.

2005

An exposition is given, partly historical and partly mathematical, of the Riemann zeta function � ( s ) and the associated Riemann hypothesis. Using techniques similar to those of Riemann, it is shown how to locate and count non-trivial zeros of � ( s ). Relevance of these investigations to the theory of the distribution of prime numbers is discussed.

Distribution of Primes , 2017

In this paper we expand the prime number theorem, twice. Then we use both expansions to describe the distribution of primes. Afterwards we analyze the symmetry in the asymptotic equivalence. Before we come to the Weil conjecture, we derive a solution of the non-trivial zeros of the Riemann zeta function, the root identity, which we expand to the entire complex plane via a replication identity and which we employ to show, that the critical strip next to the critical line is a zero free zone.

2023

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 examines the mysterious non-trivial zeros of the Riemann zeta function ζ and explains their role, e.g., in the computation of the error term in Riemann’s J function for estimating the quantity of primes less than a given number. The paper also explains the close connection between the Riemann zeta function ζ and the prime numbers. [Published in international mathematics journal.]

arXiv: General Mathematics, 2017

Our aim in this work is to present a modern study of the classical Riemann zeta function aided by contemporary computational tools. One of our main discoveries in this work regarding $\zeta(z)$ is the observation of the following natural property (alongside various other new properties): $$ \vert \zeta (0.5 + y i ) \vert < \vert \zeta \left ( x+y i \right )\vert \hspace{0.5cm} \textrm {for all } \hspace{0.25cm} 0 \leq x<0.5 \textrm{ and } 6.29<y<Y,$$ with $0<<Y$ (we conjecture that $Y=+\infty$). In particular, the Riemann hypothesis is a direct consequence of the above property (in any domain which it holds) which gives a natural explanation to the fact that $\zeta(z)$ admits no zeros in the critical strip with $Re(z) \neq 0.5$.