On a family of arithmetic series related to the M\"obius function

TL;DR

This paper extends previous results on sums involving the M"obius function and prime factors by analyzing sums over primes with a given natural density, providing convergence estimates.

Contribution

It generalizes earlier work by considering arbitrary prime sets with density, not just arithmetic progressions, and offers effective convergence rates.

Findings

Sum over primes with density involving nd (n)(n)/n converges to zero.

Provides explicit estimates for the convergence rate.

Extends results from arithmetic progressions to more general prime sets.

Abstract

Let $P^-(n)$ denote the smallest prime factor of a natural integer $n>1$. Furthermore let $\mu$ and $\omega$ denote respectively the M\"obius function and the number of distinct prime factors function. We show that, given any set ${{\scr P}}$ of prime numbers with a natural density, we have $\sum_{P^-(n)\in \scr P}\mu(n)\omega(n)/n=0$ and provide a effective estimate for the rate of convergence. This extends a recent result of Alladi and Johnson, who considered the case when ${\scr P}$ is an arithmetic progression.