On the Dirichlet divisor problem in short intervals

TL;DR

This paper investigates the behavior of the error term in the Dirichlet divisor problem within short intervals, providing new bounds and insights into its fluctuations when the interval length is smaller than the main variable.

Contribution

It introduces novel results on the differences of the divisor problem's error term over short intervals, advancing understanding of its oscillatory nature.

Findings

New bounds for elta(x+U)-elta(x) in short intervals

Enhanced understanding of the error term's fluctuations

Results applicable for U=o(x) in divisor problem analysis

Abstract

We present several new results involving $\Delta(x+U)-\Delta(x)$, where $U = o(x)$ and $$ \Delta(x):=\sum_{n\le x}d(n)-x\log x-(2\gamma-1)x $$ is the error term in the classical Dirichlet divisor problem.