It is a consequence of some forms of the prime number theorem that with $\mu$ the Möbius function, for all $A > 0$, there exists $c_A$ such that for all sufficiently large $x$, $$\frac{1}{x}\sum_{n\le x} \mu(n)\le c_A(\log x)^{-A}.$$ Are there any papers that give explicit bounds of at least this strength?
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4$\begingroup$ I would check Tim Trudgian's work; I know that he has done explicit bounds for the usual form of the prime number theorem with an error term even stronger than this. $\endgroup$– Greg MartinSep 23, 2018 at 8:15
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$\begingroup$ The oldest paper I know with general, explicit results of this form is by El Marraki. The bounds have been improved since then, notably by Ramare. I can look up the references tomorrow. $\endgroup$– H A HelfgottSep 23, 2018 at 22:50
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