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.



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


