The question is to complete this analogy:
$$\left|Z(t)\right|=\left|\zeta \left(\frac{1}{2}+i t\right)\right| \tag{1}$$
is to:
$$Z(t)=e^{i \vartheta (t)} \zeta \left(\frac{1}{2}+i t\right) \tag{2}$$
as:
$$\left|f(t)\right|=\left|\sum\limits_{n=1}^{n=k} \frac{1}{n} \zeta(1/2+i t)\sum\limits_{d|n}\frac{\mu(d)}{d^{(1/2+i t-1)}}\right| \tag{3}$$
is to what? $f(t) = \text{?} \tag{4}$
I am asking this because $f(t)$ divided by Riemann Siegel theta and the $k$-th Harmonic number:
$$f_{\vartheta(t)}(t)=\frac{\text{sgn} (Z(t))\left|\sum\limits_{n=1}^{n=k} \frac{1}{n} \zeta(1/2+i \cdot t)\sum\limits_{d|n}\frac{\mu(d)}{d^{(1/2+i \cdot t-1)}}\right|}{g(t)+H_{\text{k}}}$$
where:
$$g(t)=\frac{\partial \vartheta (t)}{\partial t}$$
and where $\vartheta(t)$ is the Riemann-Siegel theta function,
has a nice plot:
The blue graph passes through the Riemann zeta zeros, like the Riemann-Siegel zeta function. However, the square wave $f(t)$ is divergent despite the appearance of the plot.
$Z(t)$ is the Riemann-Siegel zeta function.