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I'm searching the best known upper bound for the Mertens function, but without assuming the Riemann hypothesis.

Landau, in 1901, have proved that $M(x)= O(x \exp(-c\sqrt{\ln x})$, but I am unable to find the current estimate in the classical literature.

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    $\begingroup$ The best known zero-free region, due to Korobov and Vinogradov (see the introduction here), gives the same upper bound for $M(x)$ as the best known error term in the prime number theorem. $\endgroup$ – Greg Martin Feb 24 '18 at 16:05
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As Greg Martin said in a comment, the Korobov-Vinogradov zero-free region for $\zeta(s)$ yields $$M(x)\ll x\exp\bigl(-c(\log x)^{3/5}(\log\log x)^{-1/5}\bigr).$$ For a reference, see Satz 3 in Section V.5 of Walfisz: Weylsche Exponentialsummen in der neueren Zahlentheorie (VEB Deutscher Verlag der Wissenschaften, Berlin, 1963).

This bound cannot be improved (essentially) without improving the Korobov-Vinogradov zero-free region, see Allison: On obtaining zero-free regions for the zeta-function from estimates of $M(x)$, Proc. Cambridge Philos. Soc. 67 (1970), 333-337.

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  • $\begingroup$ Many thanks.I knew that the zero-free region of Vinogradov-Korobov had an influence on M (x) but the reference ... $\endgroup$ – Claudeh5 Feb 24 '18 at 21:13

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