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

TL;DR

This paper establishes new mean value estimates involving the divisor error term and the Riemann zeta-function, including asymptotic formulas for integrals of their powers, advancing understanding in analytic number theory.

Contribution

It provides novel bounds and asymptotic formulas for integrals involving the divisor problem error term and the zeta-function, extending previous results to higher powers.

Findings

Bounded the integral of the divisor error term times the zeta function squared by T(log T)^4.

Derived asymptotic formulas for integrals of the divisor error term raised to powers 2 to 8 times the zeta function squared.

Identified explicit constants and error terms in the asymptotic formulas for these integrals.

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 $$ \int_1^{T}\Delta^{k}(t)|\zeta(1/2+it)|^2\,dt=c_1(k)T^{1+\frac k4}\log T+ c_2(k)T^{1+\frac k4}+O_\varepsilon(T^{1+\frac k4-\eta_k+\varepsilon}), $$ 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 $|\zeta(1/2+it)|$.