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

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.