For integers $n\geq 1$, taking $k\geq 1$ for $$z_k:=\mu(k)+i,$$ where $\mu(k)$ is the Möbius function and $i=\sqrt{-1}$ the complex imaginay unit, then we define the (real) arithmetical function $$f(n)= \sum_{k=1}^{n} \left| z_k \right|^2- \left( 1+\frac{6}{\pi^2} \right) \left( \left( \sum_{k=1}^{n} z_k \right)\cdot \left( \sum_{k=1}^{n} \bar{ z}_k \right)- \sum_{k=1}^{n}\left( \frac{z_k+\bar{ z}_k}{2} \right)^2 \right) ^{\frac{1}{2}} $$ where we are denoting the complex conjugate of a complex number with $\bar{ z}$.
Question. I'm not looking a simplification of such arithmetical function, neither its properties. Those what I am asking is if you can deduce an interpretation, or you known it from the literature, without a simplification of such arithmetical function (see my Attempts). Thanks in advance.
I am interested, if the (arithmetical function) $(z_k)_{k\geq 1}$ was in the literature, looking to get closed form as previous arithmetical function $f(n)$. I am interested in such $f(n)$, since in the literature there is a condition (an asymptotic identity) involving, now yes, its simplified form (but I don't can provide us an open acess of such condition, and this isn't myself question).
If it isn't possible, I understand that myself is a speculative way (in my attempts my claims I am asking to me about an explanation that perhaps there no exists and about the importance...), and there aren't answers I should delete my post in the next weeks.
Attempts. Each one of my attempts were uncompleted, I am looking an explanation of why the artificious defined $f(n)$ could be useful to study the distribution of the values of $\mu(n)$. We have also that $f(n)$ is positive and unbounded (but it isn't the explanation or relation that I am looking; I am looking a special meaning of such $f(n)$).
The first one (I hope that my few words have sense), is that we claim that $\mathbb{C}$ is isomorphic with the plane $ \mathbb{R}^2 $, and the lattice points (the Möbius function always take integers values), and I am looking an explanation or notion that provide us the importance of the distribution of $(z_k)_{k\geq 1}$ as subset of the lattice. And we have that $\frac{6}{\pi^2}$ is the density of the lattice points that are visible from the origin, and
$$\frac{6}{\pi^2}=\sum_{k=1}^{\infty}\frac{\mu(k)}{k^2}:=\sum_{k=1}^{\infty}\frac{z_k+\bar{ z}_k}{2k^2},$$
but it doesn't say nothing to me about why the shape of the function $f(n)$ is special, from the distribution of $(z_k)_{k\geq 1}$ as subset of the lattice.
My second attempt was think about statistics, where the squares, sum of squares, differences and the square root are importants to analyze a (random) sample $(z_k)_{k=1}^n$ (in this point I known, and I hope that my words are right that statistics is based in the theory of $\mathcal{l}^2(F)$ spaces, thus we are working with one of them, over the field $F= \mathbb{C} $ ). I knonw about the variance, and other definitions, but I don't know if I can get a explanation from those. Since in this attempt the squares have importance, I can write this time, $\frac{6}{\pi^2}$ as $\frac{1}{\zeta(2)}$, where as we know $\zeta(2)=\sum_{k=1}^{\infty}\frac{1}{k^2}$. And perhaps it is possible write $1$, if such interpretation has mathematical meaning, using the sample $z_k$, with $\Re$, $\Im$ or the complex conjugancy.