Almost periodicity of $C(e^u) = \sum_{n \le e^u} \lambda(n)/n$ under $x = e^u$

Let $\lambda(n) = (-1)^{\Omega(n)}$ be the Liouville function and $$ C(x) = \sum_{n \le x} \frac{\lambda(n)}{n}. $$

By Turán's classical approach, $C(x)$ is controlled by $$ \sum_{n=1}^\infty \frac{\lambda(n)}{n^{s+1}} = \frac{\zeta(2s+2)}{\zeta(s+1)}, $$ and each nontrivial zero $\rho = \beta + i\gamma$ of $\zeta$ contributes a term of size $x^{\beta-1}$ to $C(x)$.

Setting $u = \log x$ and $G(u) := C(e^u)$, each such term becomes $$ e^{(\beta-1)u} \cos(\gamma u + \phi_\rho), $$ i.e. a damped pure frequency in $u$.

Has $G(u)$ been rigorously studied as an almost periodic function of $u$, with frequency set $\{\gamma : \zeta(\tfrac12+i\gamma)=0\}$? In particular, are there known bounds on $$ \frac{1}{U}\int_0^U |G(u)|^2\, du $$ obtained via this viewpoint? References would be appreciated.