相关论文: On the Riemann zeta-function and the divisor probl…

A discussion involving the evaluation of the sum $$\sum_{T<\g\le T+H}|\zeta(1/2+i\gamma)|^2$$ and some related integrals is presented, where $\gamma\,(>0)$ denotes imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. It…

This paper is closely related to the recent work [BW17] of the same authors and our purpose is to elaborate more on some of the results and methods from [BW17]. More specifically our goal is two-fold. Firstly, we will indicate how a simple…

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…

Assuming the Riemann Hypothesis we obtain an upper bound for the moments of the Riemann zeta-function on the critical line. Our bound is nearly as sharp as the conjectured asymptotic formulae for these moments. The method extends to moments…

Power moments of $$ J_k(t,G) = {1\over\sqrt{\pi}G} \int_{-\infty}^\infty |\zeta(1/2 + it + iu)|^{2k}{\rm e}^{-(u/G)^2} du \qquad(t \asymp T, T^\epsilon \le G \ll T),$$ where $k$ is a natural number, are investigated. The results that are…

An explicit estimate for the Riemann zeta function on the critical line is derived using the van der Corput method. An explicit van der Corput lemma is presented.

Mean square estimates for $Z_2(s) = \int_1^\infty |\zeta(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are discussed, and some related Mellin transforms of quantities connected with the fourth power moment of $|\zeta(1/2+ix)|^4$.

Andrade and Keating computed the mean value of quadratic Dirichlet $L$--functions at the critical point, in the hyperelliptic ensemble over a fixed finite field $\mathbb{F}_q$. Summing $L(1/2,\chi_D)$ over monic, square-free polynomials $D$…

Hardy and Littlewood initiated the study of the $2k$-th moments of the Riemann zeta function on the critical line. In 1918 Hardy and Littlewood established an asymptotic formula for the second moment and in 1926 Ingham established an…

The meromorphic function $W(s)$ introduced in the Riemann-Zeta function $\zeta(s) = W(s) \zeta(1-s)$ maps the line of $s = 1/2 + it$ onto the unit circle in $W$-space. $|W(s)| = 0$ gives the trivial zeroes of the Riemann-Zeta function…

Suppose $a$ and $b$ are two fixed positive integers such that $(a,b)=1.$ In this paper we shall establish an asymptotic formula for the mean square of the error term $\Delta_{a,b}(x)$ of the general two-dimensional divisor problem.

The pseudomoments of the Riemann zeta function, denoted $\mathcal{M}_k(N)$, are defined as the $2k$th integral moments of the $N$th partial sum of $\zeta(s)$ on the critical line. We improve the upper and lower bounds for the constants in…

Let $\Delta_1(x;\phi)$ be the error term of the first Riesz means of the Rankin-Selberg zeta function. We study the higher power moments of $\Delta_1(x;\phi)$ and derive an asymptotic formula for 3-rd, 4-th and 5-th power moments by using…

We evaluate the first moment of central values of the family of quadratic Dirichlet $L$-functions using the method of double Dirichlet series. Under the generalized Riemann hypothesis, we prove an asymptotic formula with an error term of…

Explicit bounds on the tails of the zeta function $\zeta$ are needed for applications, notably for integrals involving $\zeta$ on vertical lines or other paths going to infinity. Here we bound weighted $L^2$ norms of tails of $\zeta$. Two…

In order to well understand the behaviour of the Riemann zeta function inside the critical strip, we show; among other things, the Fourier expansion of the $\zeta^k(s)$ ($k \in \mathbb{N}$) in the half-plane $\Re s > 1/2$ and we deduce a…

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)…

Let $\mathop{\mathcal R}(s)$ be the function related to $\zeta(s)$ found by Siegel in the papers of Riemann. In this paper we obtain the main terms of the mean values \[\frac{1}{T}\int_0^T |\mathop{\mathcal…

The main task of this work is to give an improvement for the upper bounds of the Laplace transform $$\int_0^{+\infty}\Bigl|\zeta\left(\frac{1}{2}+it\right)\Bigr|^{2\beta}e^{-\delta t}dt \ll_{\beta,\varepsilon}…

In this paper, we present an improved explicit subconvexity result for the Riemann zeta function $\zeta\left( s\right)$ along the critical line $s=1/2+it$, given by Hiary, Patel and Yang in 2024. This new bound is derived by combining a…