On certain integrals involving the Dirichlet divisor problem

TL;DR

This paper establishes new upper bounds for integrals involving error terms related to the Dirichlet divisor problem, improving upon previous estimates and extending results to analogous error terms in the mean square formula for the zeta function.

Contribution

It provides sharper bounds for integrals of error terms in divisor problems and their analogues, advancing understanding of their asymptotic behavior.

Findings

Bounds for (x)_3(x) and (x)_4(x) integrals are sharper than previous estimates.

New bounds are derived for error terms involving the Riemann zeta function's mean square.

Results extend to error terms in the mean square formula for |z(1/2+it)|.

Abstract

We prove that $$ \int_1^X\Delta(x)\Delta_3(x)\,dx \ll X^{13/9}\log^{10/3}X, \quad \int_1^X\Delta(x)\Delta_4(x)\,dx \ll_\varepsilon X^{25/16+\varepsilon}, $$ 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 $|\zeta(1/2+it)|$.