Higher moments of the error term in the divisor problem

TL;DR

This paper establishes bounds on higher moments of the error term in the divisor problem, using advanced formulas and bounds, and also investigates the size of the error term in moments of a specific case.

Contribution

It provides new bounds for the fourth moment of the error term in the divisor problem and analyzes the size of the error in moments of (x), advancing understanding of divisor problem error terms.

Findings

Bound on the integral of _k(x)^4 over a short interval.

Application of Voronoef-type formula and Robert--Sargos bounds.

Analysis of the size of the error term in moments of (x).

Abstract

It is proved that, if $k\ge2$ is a fixed integer and $1 \ll H \le X/2$, then $$ \int_{X-H}^{X+H}\Delta^4_k(x)\d x \ll_\epsilon X^\epsilon\Bigl(HX^{(2k-2)/k} + H^{(2k-3)/(2k+1)}X^{(8k-8)/(2k+1)}\Bigr), $$ where $\Delta_k(x)$ is the error term in the general Dirichlet divisor problem. The proof uses the Vorono{\"\i}--type formula for $\Delta_k(x)$, and the bound of Robert--Sargos for the number of integers when the difference of four $k$--th roots is small. We also investigate the size of the error term in the asymptotic formula for the $m$-th moment of $\Delta_2(x)$.