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

TL;DR

This paper derives an asymptotic formula with a specific error bound for the integral of the fourth power of the divisor problem's error term, advancing understanding of higher-power moments in number theory.

Contribution

It provides a new asymptotic formula with an explicit error term for the fourth moment of the divisor problem error term, extending previous results.

Findings

Asymptotic formula for (T^{53/28+psilon}) error for ^4(x)

Results applicable to other well-known error terms in analytic number theory

Enhanced understanding of higher-power moments of divisor problem error term

Abstract

Let $\Delta(x)$ be the error term of the Dirichlet divisor problem. An asymptotic formula with the error term $O(T^{53/28+\epsilon})$ is established for the integral $\int_1^T\Delta^4(x)dx.$ Similar results are also established for some other well-known error terms in the analytic number theory .