On some upper bounds for the zeta-function and the Dirichlet divisor problem

TL;DR

This paper establishes new upper bounds for integrals involving the error term in the Dirichlet divisor problem and the Riemann zeta-function, extending previous asymptotic results to bounds.

Contribution

It provides novel upper bounds for integrals of powers of the divisor problem error term multiplied by zeta-function powers, complementing earlier asymptotic formulas.

Findings

Derived upper bounds for integrals involving elta^k(t)|zeta(1/2+it)|^{2m}

Extended previous asymptotic results to bounds for a wider range of parameters

Enhanced understanding of the behavior of the divisor problem and zeta-function integrals

Abstract

Let $d(n)$ be the number of divisors of $n$, let $$ \Delta(x) := \sum_{n\le x}d(n) - x(\log x + 2\gamma -1) $$ denote the error term in the classical Dirichlet divisor problem, and let $\zeta(s)$ denote the Riemann zeta-function. Several upper bounds for integrals of the type $$ \int_0^T\Delta^k(t)|\zeta(1/2+it)|^{2m}dt \qquad(k,m\in\Bbb N) $$ are given. This complements the results of the paper Ivi\'c-Zhai [Indag. Math. 2015], where asymptotic formulas for $2\le k \le 8,m =1$ were established for the above integral.