2021/02/11 by Maxie D. Schmidt, Schmidt, Maxie Dion
Mathematics · Physics and Astronomy · #11A25 #11N37 #11N60 #Advanced Mathematical Identities #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT) #Quantum chaos and dynamical systems #and 11N64
paper · pdf · doi:10.48550/arxiv.2102.05842
openalex publication_date 2021/02/11 · openalex created_date 2022/07/22 · openalex updated_date 2026/07/28
The Mertens function, M(x) := ∑n ≤ x μ(n), is defined as the summatory function of the classical Möbius function. The Dirichlet inverse function g(n) := (ω+1)-1(n) is defined in terms of the shifted strongly additive function ω(n) that counts the number of distinct prime factors of n without multiplicity. The Dirichlet generating function (DGF) of g(n) is ζ(s)-1 (1+P(s))-1 for \Re(s) > 1 where P(s) = ∑p p-s is the prime zeta function. We study the distribution of the unsigned functions |g(n)| with DGF ζ(2s)-1(1-P(s))-1 and CΩ(n) with DGF (1-P(s))-1 for \Re(s) > 1. We establish formulas for the average order and variance of log CΩ(n) and prove a central limit theorem for the distribution of its values on the integers n ≤ x as x → ∞. Discrete convolutions of the partial sums of g(n) with the prime counting function provide new exact formulas for M(x).