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?