This is one of many questions that has been answered in the comments, so I will just summarize the answer with a CW posting: $M(x) < x^{1/\sqrt{e}}$ according to the poster Greg Martin. In a computer search, the ratio seems to converge to roughly $0.74$ for $x$ up to a million. On the other hand, it doesn't converge very quickly and might possibly not converge at all. All that was have from the original posting is that the lim sup of the ratio is finite (and at most 1 according to the answer) while the lim inf is positive.
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