On the fourth power moment of $\Delta(x)$ and $E(x)$ in short intervals

TL;DR

This paper investigates the fourth power moments of the error terms in divisor sum and zeta function mean square within short intervals, providing asymptotic formulas for specific ranges.

Contribution

It derives new asymptotic formulas for the fourth power moments of $ riangle(x)$ and $E(x)$ in short intervals, extending previous understanding of their behavior.

Findings

Asymptotic formulas for the fourth power moments are established.

Results apply to short intervals of Jutila's type.

The work advances knowledge of error term fluctuations in number theory.

Abstract

Let $\Delta(x)$ and $E(x)$ be error terms of the sum of divisor function and the mean square of the Riemann zeta function, respectively. In this paper their fourth power moments for short intervals of Jutila's type are considered. We get an asymptotic formula for $U$ in some range.