Conversions explicites des nombres premiers vers la fonction de Möbius

We prove that $\mathrm{\mid }{\sum }_{n\le x}\mu (n)\mathrm{\mid }\le x\mathrm{/}160383$ for all $x\ge 8.4\times {10}^9$ thus improving the record for this type of estimates which was $\mathrm{\mid }{\sum }_{n\le x}\mu (n)\mathrm{\mid }\le x\mathrm{/}4345$ for all $x\ge 2160535$ and was due to Cohen, Dress and El Marraki. We also prove that $\mathrm{\mid }{\sum }_{n\le x}\mu (n)\mathrm{\mid }\le x\mathrm{/}180194$ for all $x\ge {10}^{19}$.