Related papers: Zeros of the Riemann Zeta Function

In the present paper, we show that under the Riemann hypothesis, and for fixed $h, \epsilon > 0$, the supremum of the real and the imaginary parts of $\log \zeta (1/2 + it)$ for $t \in [UT -h, UT + h]$ are in the interval $[(1-\epsilon)…

This paper studies the non-holomorphic Eisenstein series E(z,s) for the modular surface, and shows that integration with respect to certain non-negative measures gives meromorphic functions of s that have all their zeros on the critical…

In this article, we show that the Riemann hypothesis for an $L$-function $F$ belonging to the Selberg class implies that all the derivatives of $F$ can have at most finitely many zeros on the left of the critical line with imaginary part…

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…

Assuming the Riemann hypothesis, we obtain upper and lower bounds for moments of the Riemann zeta-function averaged over the extreme values between its zeros on the critical line. Our bounds are very nearly the same order of magnitude. The…

The goal of this paper is to give a relatively simple proof of some known zero density estimates for Riemann zeta function which are sufficiently strong to break the density hypothesis in a nontrivial part of the critical strip. Apart from…

Assuming the Riemann Hypothesis, we improve on previous results by proving there are infinitely many zeros of the Riemann zeta-function whose differences are smaller than 0.50412 times the average spacing. To obtain this result, we…

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

Assume the Riemann hypothesis. On the right-hand side of the critical strip, we obtain an asymptotic formula for the discrete mean square of the Riemann zeta-function over imaginary parts of its zeros.

We have looked at the evaluation of the Riemann Zeta function at odd arguments and have provided a simple formula to approximate the value with exponential convergence. We have compared it with various other formulae present in literature.…

Assuming the Riemann Hypothesis, we provide explicit upper bounds for moduli of $S(t)$, $S_1(t)$, and $\zeta\left(1/2+\mathrm{i}t\right)$ while comparing them with recently proven unconditional ones. As a corollary we obtain a conditional…

On the assumption of the Riemann hypothesis, we generalize a central limit theorem of Fujii regarding the number of zeroes of Riemann's zeta function that lie in a mesoscopic interval. The result mirrors results of Soshnikov and others in…

We give a short proof of Levinson's result that more than 1/3 of the zeros of the zeta function are on the critical line.

Montgomery in 1973 introduced the pair correlation method to study the vertical distribution of Riemann zeta-function zeros. This work assumed the Riemann Hypothesis (RH). One striking application was a short proof that at least 2/3 of…

Assuming the Riemann Hypothesis, we prove that $$ N_1(T) = \frac{T}{2\pi}\log \frac{T}{4\pi e} + O\bigg(\frac{\log T}{\log\log T}\bigg), $$ where $N_1(T)$ is the number of zeros of $\zeta'(s)$ in the region $0<\Im s\le T$.

The Dirichlet eta function can be divided into $n$-th partial sum $\eta_{n}(s)$ and remainder term $R_{n}(s)$. We focus on the remainder term which can be approximated by the expression for $n$. And then, to increase reliability, we make…

Using the $\zeta$ functional equation and the Hadamard product, an analytical expression for the sum of the reciprocal of the $\zeta$ zeros is established. We then demonstrate that on the critical line, $|\zeta|$ is convex, and that in the…

We consider the alternating Riemann zeta function $\zeta^*(s)= \sum^{\infty} _{ n=1} \frac{(-1)^{n-1}}{n^s}$, which converges if $Re (s)>0 .$ By using Rouche's theorem, the Bolzano-Weierstrass theorem and by method of contradiction we…

Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann zeta function, in particular the zeros of this function. Theorem 1 gives a zero-free region.…

We investigate the distribution of the zeros of partial sums of the Riemann zeta-function, sum_{n\leq X}n^{-s}, estimating the number of zeros up to height T, the number of zeros to the right of a given vertical line, and other aspects of…