Consider the function $a(n)$ defined in formula (1) below and it's summatory function $f(x)$ defined in formula (2) below where $f(x)$ is related to the Riemann zeta function $\zeta(s)$ as illustrated in formula (3) below.
$$a(n)=\sum\limits_{d|n}\mu(d)\ \mu\left(\frac{n}{d}\right)\tag{1}$$
$$f(x)=\sum\limits_{n=1}^x a(n)\tag{2}$$
$$F(s)=s\int_0^\infty f(x)\ x^{-s-1}\ ds=\underset{N\to\infty}{\text{lim}}\left(\sum_{n=1}^N a(n)\ n^{-s}\right)=\frac{1}{\zeta(s)^2}\ ,\quad\Re(s)>1\tag{3}$$
The following list illustrates the values taken on by $f(x)$ at the first $100$ positive integer values of $x$. Note that $f(x)$ jumps around more than Mertens function $M(x)=\sum\limits_{n=1}^x \mu(n)$ which only ever takes a step of $\pm 1$.
$\{1,-1,-3,-2,-4,0,-2,-2,-1,3,1,-1,-3,1,5,5,3,1,-1,-3,1,5,3,3,4,8,8,6,4,-4,-6,-6,-2,2,6,7,5,9,13,13,11,3,1,-1,-3,1,-1,-1,0,-2,2,0,-2,-2,2,2,6,10,8,12,10,14,12,12,16,8,6,4,8,0,-2,-2,-4,0,-2,-4,0,-8,-10,-10,-10,-6,-8,-4,0,4,8,8,6,10,14,12,16,20,24,24,22,20,18,19\}$
Figure (1) below illustrates a discrete plot of the number of times $f(x)=k$ when $x$ is an integer in the range $0<x\le 10000$. In Figure (1) below, $k$ is the horizontal axis and the number of times $f(x)=k$ is the vertical axis.
Figure (1): Illustration of counts of $f(x)=k$ when $0<x\le 10000$
Assuming $x\in\mathbb{Z}_{>0}$, I'm wondering if $f(x)=k$ an infinite number of times for every integer $k$ as $x\to\infty$. I'm particularly interested in whether $f(x)=0$ and $f(x)=4$ an infinite number of times as $x\to\infty$ (for reasons that will become apparent in a question I'm planning on posting in the near future).
Question: Is it true that $f(x)=k$ at an infinite number of positive integers $x$ for every integer $k$? If this is too hard a question, is it true that $f(x)=0$ and $f(x)=4$ both at an infinite number of positive integers $x$?