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The Mertens function $M(x)$ is the summatory Möbius function i.e.

$$M(x) = \sum_{k=1}^{x} \mu (k)$$

The conjecture that $M(x) = \mathcal{O}\left(x^{\frac{1}{2} + \epsilon}\right)$ was shown to be equivalent to the Riemann hypothesis but a stronger version of the same was disproved by Odlyzko and te Reile in 1985. Are there any known bounds on or approximations to the Mertens function?

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The best known bound under the Riemann Hypothesis is due to Soundararajan, see here. This paper and its MathSciNet review (MR2542220) contain further information about what is known unconditionally and what is expected to be the precise order of magnitude.

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  • $\begingroup$ Exactly what I was looking for. Thank you! $\endgroup$
    – user75842
    Commented Jul 8, 2015 at 22:53
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This is discussed at considerable length in the very nice paper by Kotnik and van de Lune (in Experimental Mathematics).

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