Related papers: Some remarks on the differences between ordinates …

Let $\gamma$ denote imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Certain sums over the $\gamma$'s are evaluated, by using the function $G(s) = \sum_{\gamma>0}\gamma^{-s}$ and other techniques. Some integrals…

Let $0<\gamma_1\leq \gamma_2 \leq \cdots $ denote the ordinates of nontrivial zeros of the Riemann zeta function with positive imaginary parts. For $c>0$ fixed (but possibly small), $T$ large, and $\gamma_n\leq T$, we call a gap…

Assume the Riemann Hypothesis, and let $\gamma^+>\gamma>0$ be ordinates of two consecutive zeros of $\zeta(s)$. It is shown that if $\gamma^+-\gamma < v/ \log \gamma $ with $v<c$ for some absolute positive constant $c$, then the box $$…

Let $y\ne 0$ and $C>0$. Under the Riemann Hypothesis, there is a number $T_*>0$ $($depending on $y$ and $C)$ such that for every $T\ge T_*$, both \[ \zeta(\tfrac12+i\gamma)=0 \quad\text{and}\quad\zeta(\tfrac12+i(\gamma+y))\ne 0 \] hold for…

We provide conditional and unconditional asymptotic formulae for the exponential sums $\sum_\gamma\,\gamma^{-i\tau}$, where the summation is over the ordinates of the nontrivial zeros $\rho=\beta+i\gamma$ of the Riemann zeta-function. In…

On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros $\frac12+i\gamma$ of the Riemann zeta function, we show that the sequence \[ \Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad…

A discussion involving the evaluation of the sum $\sum_{0<\gamma\le T} |\zeta(1/2+i\gamma)|^2$ is presented, where $\gamma$ denotes imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Three theorems involving certain…

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 consider sums of the form $\sum \phi(\gamma)$, where $\phi$ is a given function, and $\gamma$ ranges over the ordinates of nontrivial zeros of the Riemann zeta-function in a given interval. We show how the numerical estimation of such…

Let $Z(t)$ be the classical Hardy function in the theory of the Riemann zeta-function. The main result in this paper is that if the Riemann hypothesis is true then for any positive integer $n$ there exists a $t_{n}>0$ such that for…

Let $\gamma$ denote the imaginary parts of complex zeros $\rho = \beta+i\gamma$ of $\zeta(s)$. The problem of analytic continuation of the function $G(s) := \sum\limits_{\gamma > 0}\gamma^{-s}$ to the left of the line $\Re s = -1$ is…

In this note, we give some explicit upper and lower bounds for the summation $\sum_{0<\gamma\leq T}\frac{1}{\gamma}$, where $\gamma$ is the imaginary part of nontrivial zeros $\rho=\beta+i\gamma$ of $\zeta(s)$.

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=…

It is proved that if $T$ is sufficiently large, then uniformly for all positive integers $\ell \leqslant (\log T) / (\log_2 T)$, we have \begin{equation*} \max_{T\leqslant t\leqslant 2T}\left|\zeta^{(\ell)}\Big(1+it\Big)\right| \geqslant…

We present an explicit formula for a weighted sum over the zeros of the Riemann zeta function. This weighted sum is evaluated in terms of a sum over the prime numbers, weighted with help of the Hermite polynomials. From the explicit formula…

In this paper we provide a new series representation for the values of Riemann zeta function at integer arguments, namely: $ \zeta(m)=\sum_{n=1}^{\infty}\frac{m(-1)^{n-1}\Gamma(1-\omega_{m}n)...\Gamma(1-\omega_{m}^{m-1}n)}{n!n^m}$, where…

Several results are obtained concerning multiplicities of zeros of the Riemann zeta-function $\zeta(s)$. They include upper bounds for multiplicities, showing that zeros with large multiplicities have to lie to the left of the line $\sigma…

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…

For any $\sigma$ with $0\leq \sigma\leq 1$ and any $T>10$ sufficiently large, let $N_{\zeta}(\sigma,K,T)$ be the number of zeros $\rho=\beta+i\gamma$ of $\zeta_{K}(s)$ with $|\gamma|\leq T$ and $\beta\geq \sigma$ and the zero being counted…

We intimate deeper connections between the Riemann zeta and gamma functions than often reported and further derive a new formula for expressing the value of $\zeta(2n+1)$ in terms of zeta at other fractional points. This paper also…