# Best estimate of the Mertens function without assuming the Riemann Hypothesis

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.

• 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. – Greg Martin Feb 24 '18 at 16:05

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.

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