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Florian Daval

Publications and source records attributed to Florian Daval.

5 recordsLinked to original sources
arXiv2026

On a Smoothed Walfisz Divisor Problem

This work is in the spirit of our previous investigation on a smooth Dirichlet divisor problem, where we now replace the Dirichlet divisor function $\tau$ by the sum-of-divisors function $\sigma$. We prove a totally explicit asymptotic formula for the sum of $\sigma(n)$ twisted by the weight $1-x/n$, which enables us to eliminate the difficult part in the classical average order of $\sigma(n)$. As a corollary, we deduce the convergence of an integral dealing with the error term in the Walfisz divisor problem. We also provide an appendix containing the necessary explicit results derived from the mean value theorem and the Euler-Maclaurin summation formula.

math.NT↗
arXiv2026

On a Smoothed Dirichlet Divisor Problem

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.

math.NT↗
arXiv2023

Conversions explicites des nombres premiers vers la fonction de M\"obius

We prove that $|\sum_{n \leq x} \mu(n) | \leq x/160\,383$ for all $x\geq 8.4 \times 10^9$ thus improving the record for this type of estimates which was $|\sum_{n \leq x} \mu(n) | \leq x/4345$ for all $x\geq 2\,160\,535$ and was due to Cohen, Dress and El Marraki. We also prove that $|\sum_{n \leq x} \mu(n) | \leq x/180\,194$ for all $x\geq 10^{19}$.

math.NT↗