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The Mertens function, $M(x) := \sum_{n \leq x} \mu(n)$, is defined as thesummatory function of the classical M\"obius function. The Dirichlet inversefunction $g(n) := (\omega+1)^{-1}(n)$ is defined in terms of the shiftedstrongly additive function $\omega(n)$ that counts the number of distinct primefactors 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 $P(s) = \sum_pp^{-s}$ is the prime zeta function. We study the distribution of the unsignedfunctions $|g(n)|$ 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 theaverage order and variance of $\log C_{\Omega}(n)$ and prove a central limittheorem for the distribution of its values on the integers $n \leq x$ as $x\rightarrow \infty$. Discrete convolutions of the partial sums of $g(n)$ withthe prime counting function provide new exact formulas for $M(x)$.

Exact formulas for partial sums of the M\"obius function expressed by partial sums weighted by the Liouville lambda function