Specifically, I find it appealing to count only squarefree numbers having $k$ prime factors, so I define

$$\pi_k(x)=\#\{n\leq x: \omega(n)=k;\mu(n)\neq0 \}$$

and consider the generating functions

\begin{eqnarray}f(z,x)&=&\sum_{k=0}^{m(x)}\pi_k(x) z^k\\ &=&\sum_{n\leq x}|\mu(n)|z^{\omega(n)}. \end{eqnarray}

On one hand, these generating functions are polynomials in $z$ of degree $$m(x)=\max \{\omega(n): n\leq x;\mu(n)\neq 0\}\sim\log x/\log\log x.$$ On the other, they are the inverse Mellin transform of

$$F(z,s)=\prod_p1+zp^{-s}=H(z,s)\zeta^z(s)$$

where $H(z,s)$ is an analytic function of $s$ for fixed $z$ which is bounded above and away from zero in any half plane $\sigma\geq\sigma_0>1/2$.

I have checked that the roots of $f(z,x)$, as polynomials in $z$, do indeed have negative real parts for all $x\leq 10^6$.

Why would the roots have negative real parts? What would it say about $\zeta(s)$ or the numbers of $k$-almost primes less than $x$ if the roots were to have negative real parts?

On the Riemann hypothesis one would expect to find the roots---in some sense---closer to the non positive integers as $x\rightarrow\infty$ because these are the only values of $z$ for which $\zeta^z(s)$ is analytic throughout a neighbourhood of $s=1$ and, therefore, the only points at which $f(z,x)\in O(x^a)$ for some $a<1$. However, I am interested in the less-restrictive conjecture that they have negative real parts.

I am aware of the notion of 'stability' of linear translation invariant systems and in dynamical systems, and it's equivalence with the positivity of the principle minors of the associated Hurwitz matrices, Routh tables, Sylvester's criterion, etc. I find that these equivalences serve more as a test than to provide reasoning but, if you can say something in this regard, I would be pleased to hear about that too.

It appears this may be related to the fact that $\zeta^z(s)$ tends to infinity or zero as $s\rightarrow 0^+$ depending on whether $\Re z$ is positive or negative. However, for small $\Re z$ and $s=1$ the convergence is very slight, and making the distinction appears to be a tricky problem.

One can generalize this question and ask about the locations of the roots of $$f(z,x;s)=\sum_{n\leq x} \frac{|\mu(n)|z^{\omega(n)}}{n^s}.$$

Computationally, it appears that the same conjecture holds whenever $s$ is real and, moreover, there is a good reason for this to be true when $s>1$ is real because $$\lim_{x\rightarrow\infty}f(z,x;s)=F(z,s)$$ and this limit certainly does have it's zeros in the left half plane. One only needs then an approximation theorem on the distribution of the zeros of a uniformly convergent sequence whose limit has a prescribed distribution of zeros. One possible approach to the original problem might then be via partial summation $$f(z,x;s)=x^{-s}f(z,x)+s\int_{1}^{x}\frac{f(z,y)dy}{y^{s+1}}$$ but I have not had any success with this yet.