  The Mertens function, $M(x) := \sum_{n \leq x} \mu(n)$, is defined as the
summatory function of the classical M\"obius function. The Dirichlet inverse
function $g(n) := (\omega+1)^{-1}(n)$ is defined in terms of the shifted
strongly additive function $\omega(n)$ that counts the number of distinct prime
factors of $n$ without multiplicity. The Dirichlet generating function (DGF) of
$g(n)$ is $\zeta(s)^{-1} (1+P(s))^{-1}$ for $\Re(s) > 1$ where $P(s) = \sum_p
p^{-s}$ is the prime zeta function. We study the distribution of the unsigned
functions $|g(n)|$ with DGF $\zeta(2s)^{-1}(1-P(s))^{-1}$ and $C_{\Omega}(n)$
with DGF $(1-P(s))^{-1}$ for $\Re(s) > 1$. We establish formulas for the
average order and variance of $\log C_{\Omega}(n)$ and prove a central limit
theorem for the distribution of its values on the integers $n \leq x$ as $x
\rightarrow \infty$. Discrete convolutions of the partial sums of $g(n)$ with
the prime counting function provide new exact formulas for $M(x)$.
