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

Abstract

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 $M(x)$.