Picking up the partial sums of the M\"{o}bius function problem with probabilistic number theory

Abstract

We revisit several hybrid multiplicative-to-additive type functions from a recent preprint article. These functions, $g(n)$ with Dirichlet generating function (DGF) $\zeta(s)^{-1} (1+P(s))^{-1}$ for $\Re(s) > 1$ where $P(s) = \sum_p p^{-s}$ is the prime zeta function, $|g(n)| = \lambda(n) g(n)$ with DGF $\zeta(2s)^{-1}(1-P(s))^{-1}$, and $C_{\Omega}(n)$ with DGF $(1-P(s))^{-1}$. Each of these function variants are defined in terms of the additive (respectively, strongly additive) functions $\omega(n)$ and $\Omega(n)$. These two auxiliary functions are used in the prior manuscript to relate partial sums of the classical M\"{o}bius function, $\mu(n)$, to signed partial sums involving the prime counting function, $\pi(x)$, and the Liouville lambda function, $\lambda(n) := (-1)^{\Omega(n)}$. In this article, we explore summing the identities from the first manuscript using several probabilistic assumptions about the independence of the values of $\Omega(n)$ and $\mu^2(n)$ for $n \leq x$ at large $x$. We recover proofs of the limiting asymptotic growth of $|M(x)| / \sqrt{x}$ whose hypotheses promise to be substantially more attainable to make rigorous than past results from other authors relying on the Riemann Hypothesis or assumption of the linear independence of the simple, non-trivial zeros of $\zeta(s)$.