On a hybrid fourth moment involving the Riemann zeta-function

Abstract

We provide explicit ranges for $\sigma$ for which the asymptotic formula \begin{equation*} \int_0^T|ζ(1/2+it)|^4|ζ(σ+it)|^{2j}dt \;\sim\; T\sum_{k=0}^4a_{k,j}(σ)\log^k T \quad(j\in\mathbb N) \end{equation*} holds as $T\to \infty$, when $1\le j\le 6$, 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

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