On a hybrid fourth moment involving the Riemann zeta-function

TL;DR

This paper determines explicit ranges of the parameter  for which a fourth moment involving the Riemann zeta-function asymptotically behaves as predicted, improving previous results and applying to a divisor problem.

Contribution

It provides improved explicit ranges for  in the asymptotic formula of a hybrid fourth moment of the Riemann zeta-function, extending previous work.

Findings

Explicit  ranges for asymptotic formula established

Improved bounds over previous results

Application to a divisor problem included

Abstract

We provide explicit ranges for $\sigma$ for which the asymptotic formula \begin{equation*} \int_0^T|\zeta(1/2+it)|^4|\zeta(\sigma+it)|^{2j}dt \;\sim\; T\sum_{k=0}^4a_{k,j}(\sigma)\log^k T \quad(j\in\mathbb N) \end{equation*} holds as $T\rightarrow \infty$, when $1\leq j \leq 6$, where $\zeta(s)$ is the Riemann zeta-function. The obtained ranges improve on an earlier result of the authors [Annales Univ. Sci. Budapest., Sect. Comp. {\bf38}(2012), 233-244]. An application to a divisor problem is also given