On higher-power moments of $\Delta(x)$(II)

TL;DR

This paper develops a unified method to derive asymptotic formulas for higher-power moments of the error term in the Dirichlet divisor problem, extending results for powers 3 through 9 and related error terms.

Contribution

It introduces a unified approach to evaluate asymptotic formulas for moments of error terms in number theory, covering multiple well-known problems.

Findings

Asymptotic formulas established for $oxed{ ext{integrals of } riangle^k(x)}$ for $3 \leq k \leq 9$

Unified method applicable to other error terms in analytic number theory

Extension of known results to higher moments with explicit asymptotics

Abstract

Let $\Delta(x)$ be the error term of the Dirichlet divisor problem. The asymptotic formula of the integral $\int_1^T\Delta^k(x)dx$ is established for any integer $3\leq k\leq 9$ by an unified method. Similar results are also established for some other well-known error terms in the analytic number theory.