On certain integrals involving the Dirichlet divisor problem

Abstract

We prove that $${\int }_1^X\Delta (x){\Delta }_3(x)dx\ll X^{13/9}{\log }^{10/3}X,{\int }_1^X\Delta (x){\Delta }_4(x)dx{\ll }_{\epsilon }{X}^{25/16+\epsilon },$$ where ${\Delta }_k(x)$ is the error term in the asymptotic formula for the summatory function of $d_k(n)$, generated by ${\zeta }^k(s)$ (${\Delta }_2(x)\equiv \Delta (x)$). These bounds are sharper than the ones which follow by the Cauchy-Schwarz inequality and mean square results for ${\Delta }_k(x)$. We also obtain the analogues of the above bounds when $\D(x)$ is replaced by $E(x)$, the error term in the mean square formula for $|\zeta (1/2+it)|$.

21 pages