Following on from this question, I include a plot of the slightly less clear, but far simpler mathematically Mertens function against $x$ to the power of Zeta Zero 1, where the correlation between the two is fairly convincing - the largest scale changes in frequency imitating those of $x^{\rho_{1}}/e$:
It appears that $$M(x)\equiv\sum_{k=1}^{x}\mu(k)\sim \frac{\Im(x^{\rho_{1}})}{e}$$
I have not tried it yet, but am wondering whether $$\frac{1}{n}\sum_{\rho_{n=1}}^{\rho_\infty}\frac{\Im(x^{\rho_{n}})}{e}$$ (or something similar) will yield the precise fluctuations of the Mertens function.
I realise that
$$\frac{1}{\zeta(s)}=\sum_{n=0}^{\infty}\frac{\mu(n)}{n^{s}}$$
but cannot make the leap to the connection with the fluctuations of the zeros. Sorry, but I am a bit out of my depth here, and would greatly appreciate some pointers in the right direction.
Update
A plot of
$$M(x)\equiv\sum_{k=1}^{x}\mu(k)\text{ against } \bigg(\frac{\Im(x^{\rho_{1}})}{e}+\frac{\Im(x^{\rho_{2}})}{e}\bigg)/2$$
for comparison:
Update 2
It appears that to divide by $e$ is not necessary once enough Zeros have been added. A plot of
$$M(x)\equiv\sum_{k=1}^{x}\mu(k)\text{ against } \bigg(\Im(x^{\rho_{1}}+x^{\rho_{2}}+x^{\rho_{3}}+\dots x^{\rho_{9}})\bigg)/9:$$
