Related papers: A note on $r$-gaps between zeros of the Riemann ze…

As well known, the important hypothesis formulated by B.G. RIEMANN in 1859 states that all non-trivial zeroes of the Zeta function $Z(s)=\sum_{n=1}^{\infty } n^{-s}$ should fall on the Critical Line (C.L.) $Re(s)=\frac{1}{2}$.\\ Although…

We present an elementary, short and simple proof of the validity of the Lindel\"of hypothesis about the Riemann zeta-function. The obtained estimate and classical results by Bohr-Landau and Littlewood disprove Riemann's hypothesis.

The Riemann zeta function at integer arguments can be written as an infinite sum of certain hypergeometric functions and more generally the same can be done with polylogarithms, for which several zeta functions are a special case. An…

In this paper, we prove that there are more than 66.036% of zeros of the Riemann zeta-function are distinct.

We prove that for any zero $\beta'+i\gamma'$ of $\zeta'(s)$ there exists a zero $\beta+i\gamma$ of $\zeta(s)$ such that $|\gamma-\gamma'|\ll \sqrt{|\beta'-\tfrac{1}{2}|},$ and we provide some other related results.

We study the values taken by the Riemann zeta-function $\zeta$ on discrete sets. We show that infinite vertical arithmetic progressions are uniquely determined by the values of $\zeta$ taken on this set. Moreover, we prove a joint discrete…

Let $d(n)$ be the number of divisors of $n$, let $\gamma$ denote Euler's constant and $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote…

We assume the Riemann hypothesis to improve upon the rate of convergence of $(\log\log\log T)^2/\sqrt{\log\log T}$ in Selberg's central limit theorem for $\log|\zeta(1/2+it)|$ given by the author. We achieve a rate of convergence of…

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…

There are many analytic functions $U(t)$ satisfying $Z(t)=2\Re\bigl\{ e^{i\vartheta(t)}U(t)\bigr\}$. Here, we consider an entire function $\mathop{\mathcal L}(s)$ such that $U(t)=\mathop{\mathcal L}(\frac12+it)$ is one of the simplest among…

The main aim of this paper is twofold. First we generalize, in a novel way, most of the known non-vanishing results for the derivatives of the Riemann zeta function by establishing the existence of an infinite sequence of regions in the…

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…

A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values,…

We show that there is a contradiction between the Riemann's Hypothesis and some form of the theorem on the universality of the zeta function.

The Riemann zeta function $\zeta(s)$ is defined as the infinite sum $\sum_{n=1}^\infty n^{-s}$, which converges when ${\rm Re}\,s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line ${\rm Re}\,s=…

The authors describe a general approach which, in principal, should produce the correct (conjectural) formula for every even integer moment of the Riemann zeta function. They carry it out for the sixth and eigth powers; in the case of sixth…

The note is a continuation of the previous paper ``On q-analogues of Riemann's zeta'' (math.QA/980499). It contains an output of the computer program calculating the zeros of the ``sharp'' q-zeta function.

Assuming the Riemann Hypothesis, Soundararajan showed that $\displaystyle{\int_{0}^{T} \vert \zeta(1/2 + it)\vert^{2k} \ll T(\log T)^{k^2 + \epsilon}}$ . His method was used by Chandee to obtain upper bounds for shifted moments of the…

It is shown explicitly how the sign of Hardy's function $Z(t)$ depends on the parity of the zero-counting function $N(T)$. Two existing definitions of this function are analyzed, and some related problems are discussed.

The Landau--Gonek Theorem evaluates $X^\rho$ summed over the non-trivial zeros of the Riemann zeta function. Their result shows great sensitivity to the arithmetic nature of $X$. We prove a related result concerning the sum of $\chi(\rho)…