On the distribution of the error terms in the divisor and circle problems

TL;DR

This paper investigates the distribution of error terms in classical problems of analytic number theory, providing bounds, shape analysis of large deviations, and extending results to related problems like the Gauss circle problem and the Riemann zeta function.

Contribution

It improves bounds on the distribution discrepancy of the divisor problem's error term and analyzes its large deviations, extending the analysis to related number theory problems.

Findings

Bounded the discrepancy between the distribution of $ riangle(x)$ and a probabilistic model.

Determined the shape of large deviations for the divisor problem's error term.

Extended results to the Gauss circle problem and the second moment of the Riemann zeta function.

Abstract

We study the distribution functions of several classical error terms in analytic number theory, focusing on the remainder term in the Dirichlet divisor problem $\Delta(x)$. We first bound the discrepancy between the distribution function of $\Delta(x)$ and that of a corresponding probabilistic random model, improving results of Heath-Brown and Lau. We then determine the shape of its large deviations in a certain uniform range, which we believe to be the limit of our method, given our current knowledge about the linear relations among the $\sqrt{n}$ for square-free positive integers $n$. Finally, we obtain similar results for the error terms in the Gauss circle problem and in the second moment of the Riemann zeta function on the critical line.