Related papers: An analytic approach to the Riemann hypothesis

The paper describes a method for calculating values of Riemann's Zeta function within the critical strip 0< {\sigma} <1 and on its boundary. The approach is based on the "Alternating Zeta function" {\eta}(s). The actual Riemann Zeta…

This analysis which uses new mathematical methods aims at proving the Riemann hypothesis and figuring out an approximate base for imaginary non-trivial zeros of zeta function at very large numbers, in order to determine the path that those…

We investigate the intersections of the curve $\mathbb{R}\ni t\mapsto \zeta({1\over 2}+it)$ with the real axis. We show that if the Riemann hypothesis is true, the mean-value of those real values exists and is equal to 1. Moreover, we show…

Two identities extracted from the literature are coupled to obtain an integral equation for Riemann's $\xi(s)$ function, and thus $\zeta(s)$ indirectly. The equation has a number of simple properties from which useful derivations flow, the…

We prove the Riemann Hypothesis via an analytically regulated surface integral over the critical strip of the Riemann zeta function. The key idea is that the convergence of this normalized integral is equivalent to the condition that all…

The Riemann Hypothesis is not proved yet and this article gives a possible proof for the hypothesis which confirms that the only possible nontrivial zeros of the Riemann zeta-function has its real value equal to 1/2. From the result, the…

it is proved that at least 41.28% zeros of the Riemann zeta function are on the critical line

We have dealt with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and showed some evidence to indicate the hypothesis. We briefly…

By using new power inequalities we give an elementary proof of an important relation for the Riemann zeta-function |\zeta(1-s)| <= |\zeta(s)| in the strip 0< Re s<1/2,\ |\Im s| >= 12. Moreover, we establish a sufficient condition of the…

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 functions. At the `non-trivial' zeros of zeta…

The Riemann Hypothesis, originally proposed by the eminent mathematician Bernard Riemann in 1859, remains one of the most profound challenges in number theory. It posits that all non-trivial zeros of the Riemann zeta function {\zeta}(s) are…

Four propositions are considered concerning the relationship between the zeros of two combinations of the Riemann zeta function and the function itself. The first is the Riemann hypothesis, while the second relates to the zeros of a…

This paper is a summary of the general approach outlined in my previous papers toward proving the riemann hypothesis. Numerical and graphical proof of the Riemann Hypothesis is presented with analytical arguments although more work needs…

We investigate the intersections of the curve $\mathbb{R}\ni t\mapsto \zeta({1\over 2}+it)$ with the real axis. We show unconditionally that the zeta-function takes arbitrarily large positive and negative values on the critical line.

In this paper is stablished a characterization of the solutions of the equation: zeta(z) = 0. Then such a characterization is used to give a proof for Riemann is Conjecture.

We show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta…

We develop a finite-dimensional, symmetric matrix framework associated with the Riemann zeta function for complex arguments s with Real(s) unequal 1/2.

The purpose of this paper is to prove that the so-called Quasi-Riemann Hypothesis for the Zeta-function implies the Riemann Hypothesis

The meromorphic function $W(s)$ introduced in the Riemann-Zeta function $\zeta(s) = W(s) \zeta(1-s)$ maps the line of $s = 1/2 + it$ onto the unit circle in $W$-space. $|W(s)| = 0$ gives the trivial zeroes of the Riemann-Zeta function…

We prove that there exist infinitely many consecutive zeros of the Riemann zeta-function on the critical line whose gaps are greater than $3.18$ times the average spacing. Using a modification of our method, we also show that there are even…