On a Smoothed Dirichlet Divisor Problem

Olivier Bordellès; Florian Daval

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arXiv · 2601.01905

On a Smoothed Dirichlet Divisor Problem

Olivier Bordellès·Florian Daval

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Abstract

Hardy showed that $\sum_{n \ioe x}\tau(n)-x(\log x +2\gamma -1)$ is not $o(x^{1/4})$. In this article, we prove that $\sum_{n \ioe x}\tau(n)(1-\frac{x}{n})-xP(\log x)=\frac{1}{4}+O \left( \frac{\log x}{x^{1/4}} \right)$, where $P$ is a polynomial of degree 2. As a corollary, this estimate enables us to settle a conjecture surmised by Berkane, Bordell\`{e}s, and Ramar\'{e} dealing with the positivity of an integral of the error term in the Dirichlet divisor problem. All results are entirely explicit and allow us to study the proximity between the remainder of the Dirichlet divisor problem and its logarithmic version.

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Olivier Bordellès, Florian Daval. 2026-01-05. On a Smoothed Dirichlet Divisor Problem. https://arxiv.org/abs/2601.01905

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