One can define $f(x) = M(e^x)/\sqrt{e^x}$, where $M$ is Mertens function. It looks like some sort of stationary random process (yes, I know it's not random process, see plot below), namely it lives mostly in the strip $y \in [-0.5, 0.5]$, and 'frequency' of crossing x-axis looks the same in different regions of x-axis.
But it behaves not like random walk process (normalized by sqrt(time)). Its quasi-auto-correlation function $R(\tau, T, \epsilon)$ has oscillations (see plot below). This quasi-auto-correlation function $R(\tau, T, \varepsilon)$ is defined as following: $$ R(\tau, T, \varepsilon) = \mathrm{E}(f(t) \cdot f(t + \tau'))\\ \;\;where \; t \in [0, T], \; \tau' \in [\tau, \tau + \varepsilon]. $$ Here $E$ stands for "expected value", $t$ is randomly sampled from $[0, T]$, and $\tau'$ is sampled from $[\tau, \tau + \varepsilon]$. Result may depend on distribution on $[0, T]$, but numerical experiments show it does not much (~ conjecture that "random process is stationary"). Anyhow, let's use uniform distribution on $[0, T]$.
Normalized Merten's function $f(t) = M(e^t)/\sqrt{e^t}$
Auto-correlation function $R(t, T, \varepsilon)$ of $f(t)$ for $T = \log(10^7)$
Is there any known results on this quasi-auto-correlation function for this normalized Merten's function?
Namely, I wonder if limit $$\lim_{T\to \infty}R(t, T, \varepsilon)$$ exists for any $t$ and $\varepsilon$. And limit $$\lim_{\varepsilon\to 0}\lim_{T\to \infty}R(t, T, \varepsilon)$$