Questions on $f(x)=\sum\limits_{n=1}^{x}a(n)$ with an infinite number of positive integer zeros

Question

This question is related to a class of functions that meet the following conditions.

(1) $\quad f(x)=\sum\limits_{n=1}^{x}a(n)$

(2) $\quad f(x)=0$ for an infinite number of values of $x\in\mathbb{Z}^*$

(3) $\quad a(n)\in\mathbb{C}$

(4) $\quad$There are no integers $m\ge 0$ and $k\ge 1$ for which $a(n)=a(m+n+k)$ for all $n>m$.

Condition (2) above requires the function $f(x)$ to have an infinite number of positive integer zeros, and condition (4) above requires $a(n)$ to be non-periodic both initially and also after some initial number of terms.

The Mertens function defined in (5) below belongs to the class of functions which meets the conditions defined in (1) to (4) above.

(5) $\quad M(x)=\sum\limits_{n=1}^{x}\mu(n)$

Obviously when $a(n)=c\,\mu(n)$ where $c$ is a constant $f(x)$ also belongs in the class of functions which meets the conditions defined in (1) to (4) above.

I've also briefly considered but not thoroughly investigated $a(n)=c\,\mu(n+k)$ where $k\in\mathbb{Z}^*$.

Question (1): What are some other examples of functions belonging to the class of functions which meets the conditions defined in (1) to (4) above?

Question (2): Is there a general method to define or derive functions of this class?

As was pointed out in user598301's answer below, the questions above are related to a very broad class of functions. The primary motivation for this question was a desire to understand how to determine whether $f(x)$ has an infinite number of integer zeros (or more generally an infinite number of zero crossings) when $a(n)$ alternates infinitely (but perhaps somewhat randomly) between positive and negative values. I acknowledge this question was not particularly well formulated. The following question illustrates a more concrete example that is perhaps of some theoretical interest.

Questions related to Moebius Transform of Characteristic Function of the Primes

I included the Dirichlet series tag because $f(x)$ defined in (1) above is related to the Dirichlet series $F(s)$ defined in (6) below.

(6) $\quad F(s)=s\int\limits_0^\infty f(x)\,x^{-s-1} dx=\sum\limits_{n=1}^\infty\frac{a(n)}{n^s}$

Answer 1

This is just a ridiculously broad class of functions. After all, all you are asking for is an arbitrary non-periodic function $f(x)$ with infinitely many zeros. Then you let $a(n) = f(n) - f(n-1)$ and you are done.

For example, $f(n)$ could be

1. $0$ if $n$ is a prime of the form $7 \mod 2018$
2. The sum of the first $n$ digits of $\pi$ otherwise.

Unless you are much more restrictive, you certainly won't be able to say anything interesting. Already imposing that $a(nm) = a(n)a(m)$ (perhaps only for $(n,m) = 1$) would make a huge difference.