On a Smoothed Dirichlet Divisor Problem
Olivier Bordellès, Florian Daval
TL;DR
The paper resolves a smoothed version of the Dirichlet divisor problem by proving that $\sum_{n\le x} \tau(n)\left(1-\frac{x}{n}\right)-xP(\log x)$ has a sharp main term $\frac{1}{4}$ with an explicit error $O\left(\frac{\log x}{x^{1/4}}\right)$, where $P$ is a quadratic polynomial. The authors develop a framework combining Euler–Maclaurin summation, precise divisor-sum decompositions via Dirichlet’s hyperbola, and explicit generalized Chowla–Walum sum estimates, to obtain fully explicit bounds. A corollary settles a conjecture of Berkane, Bordellès, and Ramaré on the positivity of an integral of the divisor-error, and the results provide a concrete link between the standard divisor remainder $\Delta(x)$ and its logarithmic version $\delta(x)$, enabling direct comparisons. Overall, the work delivers explicit control over the proximity between the Dirichlet divisor remainder and its logarithmic analogue, with potential applications to related remainder problems and their positivity properties.
Abstract
Hardy showed that $\sum_{n \ioe x}τ(n)-x(\log x +2γ-1)$ is not $o(x^{1/4})$. In this article, we prove that $\sum_{n \ioe x}τ(n)(1-\frac{x}{n})-xP(\log x)=\frac{1}{4}+O \left( \frac{\log x}{x^{1/4}} \right)$, where $P$ is a polynomial of degree 2. As a corollary, this estimate enables us to settle a conjecture surmised by Berkane, Bordellès, and Ramaré 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.