It seems that the cumulative mean of the Mertens function is very similar in behaviour to $x$ raised to the power of the first zeta zero. I tentatively notate it as:
$$\frac{1}{x}\left(\sum_{k=1}^xM(k)\right)+2\sim\Im((\text{K}x) ^{\rho_{1}}/\text{c})$$
where $\text{c}\approx64,\ \rm{K}$ is Catalan's constant, $M(k)$ is the Mertens function of $k$, and $\rho_n$ is the nth zeta zero, and the $+2$ on the LHS is to account for the mean of the Mertens function being $\frac{1}{\zeta(0)}=-2$.
Below are plots up to $1000, 10\ 000$ and $100\ 000$.
The wavelengths are as strikingly similar for small $x$ as they are for larger $x$. It seems to me to be a little too consistent to be coincidental. Has anyone else come across any literature on the subject / or can anyone shed any light on this? Why should it fluctuate with such regularity?