Related papers: On zeros of the Riemann zeta function

We prove under RH the existence of a very large positive and negative values of the argument of the Riemann zeta function on a very short intervals.

W. Luo has investigated the distribution of zeros of the derivative of the Selberg zeta function associated to compact hyperbolic Riemann surfaces. In essence, the main results in Luo's article involve the following three points: Finiteness…

For a fixed integer $k\ge 3$ and fixed $1/2 < \sigma > 1$ we consider $$ \int_1^T |\zeta(\sigma + it)|^{2k}dt = \sum_{n=1}^\infty d_k^2(n)n^{-2\sigma}T + R(k,\sigma;T), $$ where $R(k,\sigma;T) = o(T) (T\to\infty)$ is the error term in the…

Based on an equivalence relation that was established recently on exponential sums, in this paper we study the class of functions that are equivalent to the Riemann zeta function in the half-strip $\{s\in\mathbb{C}:\operatorname{Re}s>1\}$.…

A proof of the Riemann's hypothesis (RH) about the non-trivial zeros of the Riemann zeta-function is presented. It is based on the construction of an infinite family of operators D^{(k,l)} in one dimension, and their respective…

We offer two new Mellin transform evaluations for the Riemann zeta function in the region $0<\Re(s)<1.$ Some discussion is offered in the way of evaluating some further Fourier integrals involving the Riemann xi function.

A proof of the Riemann hypothesis using the reflection principle is presented.

Starting from the symmetrical reflection functional equation of the zeta function, we have found that the sigma values satisfying zeta(s) = 0 must also satisfy both |zeta(s)| = |zeta(1 - s)| and |gamma(s/2)zeta(s)| = |gamma((1 - s)/2)zeta(1…

This paper investigates the analytic properties of the Liouville function's Dirichlet series that obtains from the function F(s)= zeta(2s)/zeta(s), where s is a complex variable and zeta(s) is the Riemann zeta function. The paper employs a…

In this paper we treat the classical Riemann zeta function as a function of three variables: one is the usual complex $\adyn$-dimensional, customly denoted as $s$, another two are complex infinite dimensional, we denote it as $\b =…

We settle a conjecture of Farmer and Ki in a stronger form. Roughly speaking we show that there is a positive proportion of small gaps between consecutive zeros of the zeta-function $\zeta(s)$ if and only if there is a positive proportion…

We revisit a representation for the Riemann zeta function $\zeta(s)$ expressed in terms of normalised incomplete gamma functions given by the author and S. Cang in Methods Appl. Anal. {\bf 4} (1997) 449--470. Use of the uniform asymptotics…

We prove that any non-zero complex values $z_1,\ldots,z_n$ can be approximated by the following integral shifts of the Riemann zeta-function $\zeta(s+id_1\tau),\ldots,\zeta(s+id_n\tau)$ for infinitely many $\tau$, provided…

For $N \in \mathbb{N}$ consider the $N$-th section of the approximate functional equation $$ \zeta_N(s)= \sum_{n =1 }^N B_n(s),$$ where $$ B_n(s)= \frac{1}{2} \left [ n^{-s} + \chi(s) \cdot n^{s-1} \right ].$$ Our aim in this work is to…

The distribution of the zeros of the Euler double zeta-function $\zeta_2(s_1,s_2)$, in the case when $s_1=s_2$, is studied numerically. Some similarity to the distribution of the zeros of Hurwitz zeta-functions is observed.

We study the horizontal distribution of zeros of $\zeta'(s)$ which are denoted as $\rho'=\beta'+i\gamma'$. We assume the Riemann hypothesis which implies $\beta'\geqslant1/2$ for any non-real zero $\rho'$, equality being possible only at a…

It is well-known that upper bounds for moments of the Riemann zeta function $\zeta(s)$ have implications for subconvexity bounds. In this paper we explore some implications in the opposite direction using functional analysis in the…

If $0 < \gamma_1 \le \gamma_2 \le \gamma_3 \le \ldots$ denote ordinates of complex zeros of the Riemann zeta-function $\zeta(s)$, then several results involving the maximal order of $\gamma_{n+1}-\gamma_n$ and the sum $$ \sum_{0<\gamma_n\le…

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.

We consider the real part $\Re(\zeta(s))$ of the Riemann zeta-function $\zeta(s)$ in the half-plane $\Re(s) \ge 1$. We show how to compute accurately the constant $\sigma_0 = 1.19\ldots$ which is defined to be the supremum of $\sigma$ such…