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

TL;DR

This paper derives exact formulas for partial sums of the Möbius function using weighted sums involving the Liouville function, and explores their distribution and average properties.

Contribution

It introduces new exact formulas for the Möbius partial sums expressed through convolutions with the Liouville function, expanding understanding of their distribution.

Findings

Formulas for the average order and variance of g(n) and C_{\u03a9}(n).

Proves a central limit theorem for the distribution of C_{}(n) on integers.

Provides new exact formulas for the Mertens function via convolutions of partial sums.

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 $M(x)$.