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

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…

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 $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $s=\tfrac12+it$. Assuming the Riemann hypothesis, we give a new and simple proof of the sharpest known bound for $S(t)$. We discuss a generalization of this bound…

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 $$…

This article improves the estimate of $|S_1(t_2)-S_1(t_1)|$, which is the definite integral of the argument of the Riemann zeta-function between $t_1$ and $t_2$. Estimates of this quantity are needed to apply Turing's method to compute the…

We show that the generalized Riemann hypothesis implies that there are infinitely many consecutive zeros of the Riemann zeta function whose spacing is 2.9125 times larger than the average spacing. This is deduced from the calculation of the…

We find an asymptotic expansion of Selberg's central limit theorem for the Riemann zeta function on $\sigma = \frac12 + ( \log T)^{-\theta}$ and $t \in [T, 2T]$, where $ 0 < \theta < \frac12$ is a constant.

We prove an analogue of Selberg's explicit formula for Motohashi's product (see arXiv:1104.1358v3 [math.NT]). We also provide a zero-density theorem for the product, which follows from Soundararajan's theorem for moments of the Riemann…

We investigate the distribution of the Riemann zeta-function on the line $\Re(s)=\sigma$. For $\tfrac 12 < \sigma \le 1$ we obtain an upper bound on the discrepancy between the distribution of $\zeta(s)$ and that of its random model,…

Using the twisted fourth moment of the Riemann zeta-function we study large gaps between consecutive zeros of the derivatives of Hardy's function $Z(t)$, improving upon previous results of Conrey and Ghosh [J. London Math. Soc. 32 (1985),…

Assuming the Riemann Hypothesis we show that there exist infinitely many consecutive zeros of the Riemann zeta-function whose gaps are greater than 2.9 times the average spacing.

Small values of $|\zeta(1/2+it)|$ are investigated, using the value distribution results of A. Selberg. This gives an asymptotic formula for $\mu(\{0 < t \le T : |\zeta(1/2+it)| \le c\})$. Some related problems involving values of…

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…

In this paper, we derive new lower bounds for the normalized distances between consecutive maxima of the Riemann zeta-function on the critical line subject to the truth of the Riemann hypothesis. The method of our proofs relies on a Sobolev…

The Riemann hypothesis, stating that the real part of all non-trivial zero points fo the zeta function must be $\frac{1}{2}$, is one of the most important unproven hypothesises in number theory. In this paper we will proof the Riemann…

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…

Let $N(\sigma,T)$ denote the number of nontrivial zeros of the Riemann zeta function with real part greater than $\sigma$ and imaginary part between $0$ and $T$. We provide explicit upper bounds for $N(\sigma,T)$ commonly referred to as a…

Let $S(\sigma,t)=\frac{1}{\pi}\arg\zeta(\sigma+it)$ be the argument of the Riemann zeta-function at the point $\sigma+it$ in the critical strip. For $n\geq 1$ and $t>0$, we define \begin{equation*} S_{n}(\sigma,t) = \int_0^t…

An explicit identity of sums of powers of complex functions presented via this a closed-form formula of Riemann zeta function produced at any given non-zero complex numbers. The closed-form formula showed us Riemann zeta function has no…

The research shows that Riemann proved that all of zeros of Riemann's zeta function are on $\sigma=1/2$ based on the functional equation \begin{align*} \pi^{-\frac{s}{2}}\Gamma \left( \frac{s}{2} \right) \zeta(s)&={\frac{1}{s(s-1)} +…