On a Smoothed Dirichlet Divisor Problem
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912817041375232 |
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| author | Bordellès, Olivier Daval, Florian |
| author_facet | Bordellès, Olivier Daval, Florian |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01905 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On a Smoothed Dirichlet Divisor Problem Bordellès, Olivier Daval, Florian Number Theory 11A25, 11L07 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. |
| title | On a Smoothed Dirichlet Divisor Problem |
| topic | Number Theory 11A25, 11L07 |
| url | https://arxiv.org/abs/2601.01905 |