On some mean value results for the zeta-function and a divisor problem II

Abstract

Let $d(n)$ be the number of divisors of $n$, let $\gamma$ denote Euler's constant and $$\Delta (x):=\sum _{n\le x}d(n)-x(\log x+2\gamma -1)$$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta (s)$ denote the Riemann zeta-function. It is shown that $${\int }_0^T\Delta (t)|\zeta (1/2+it){|}^{2}dt\ll T(\log T{)}^4.$$ Further, if $2\le k\le 8$ is a fixed integer, then we prove the asymptotic formula $${\int }_1^T{\Delta }^k(t)|\zeta (1/2+it){|}^{2}dt=c_1(k)T^{1+\frac{k}{4}}\log T+c_2(k)T^{1+\frac{k}{4}}+O_{\epsilon }(T^{1+\frac{k}{4}-{\eta }_k+\epsilon }),$$ where $c_1(k)$ and $c_2(k)$ are explicit constants, and where $${\eta }_2=3/20,{\eta }_3={\eta }_4=1/10,{\eta }_5=3/80,{\eta }_6=35/4742,{\eta }_7=17/6312,{\eta }_8=8/9433.$$ The results depend on the power moments of $\Delta (t)$ and $E(T)$, the classical error term in the asymptotic formula for the mean square of $|\zeta (1/2+it)|$.

arXiv comment31 pages

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• On some mean value results for the zeta-function and a divisor problem II