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

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 $x\geq 10^{19}$.