If the Riemann hypothesis holds, then the Mertens function $M(n)\equiv\sum_{x\leq n} \mu(n)$ behaves much like a 1D random walk insofar as

1. Every integer appears in the sequence $M(n)$ infinitely often
2. $M(n)=O(\sqrt{n})$ (ignoring subleading logarithmic corrections).

The Merten's function can be extended to the reals through an integral expression

$$M(x) = \int^{c+\mathrm{i}\infty}_{c-\mathrm{i}\infty } \frac{ds}{2\pi \mathrm{i}} \, \frac{x^s}{s \zeta(s)}\,\,\,\,\, (*)$$

My question is: Does anyone know of a smooth functions $f:\mathbb{R}\rightarrow \mathbb{R}$, which has an expression in terms of known analytic functions (e.g., Eq. $(*)$) and which also obeys condition 1. and 2. above? Perhaps there are many examples; if so, what's the simplest?