On a Smoothed Dirichlet Divisor Problem

TL;DR

This paper refines the understanding of the Dirichlet divisor problem by providing an explicit asymptotic formula for a smoothed sum involving the divisor function, settling a related conjecture and analyzing the error term's behavior.

Contribution

It introduces a new explicit asymptotic estimate for a smoothed divisor sum, confirming a conjecture about the positivity of an associated integral.

Findings

Established an explicit asymptotic formula for the smoothed divisor sum.

Confirmed the positivity conjecture for the integral of the error term.

Analyzed the proximity between the divisor problem's remainder and its logarithmic version.

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.