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On a hybrid fourth moment involving the Riemann zeta-function (1305.2685v1)
Published 13 May 2013 in math.NT
Abstract: We provide explicit ranges for $\sigma$ for which the asymptotic formula \begin{equation*} \int_0T|\zeta(1/2+it)|4|\zeta(\sigma+it)|{2j}dt \;\sim\; T\sum_{k=0}4a_{k,j}(\sigma)\logk 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