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

TL;DR

This paper improves bounds on the summatory Möbius function, establishing tighter estimates for large x, which enhances understanding of prime number distributions and Möbius function behavior.

Contribution

It provides significantly improved explicit bounds for the summatory Möbius function for large x, refining previous estimates and extending the range of validity.

Findings

Proves that |∑_{n ≤ x} μ(n)| ≤ x/160,383 for x ≥ 8.4×10^9

Establishes |∑_{n ≤ x} μ(n)| ≤ x/180,194 for x ≥ 10^19

Improves previous bounds by Cohen, Dress, and El Marraki

Abstract

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 $x\geq 10^{19}$.