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

Youness Lamzouri

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arXiv · 2410.03652

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

Youness Lamzouri

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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.

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Youness Lamzouri. 2024-10-04. On the distribution of the error terms in the divisor and circle problems. https://arxiv.org/abs/2410.03652

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