On a family of arithmetic series related to the Möbius function

Abstract

Let $P^{-}(n)$ denote the smallest prime factor of a natural integer $n>1$. Furthermore let $\mu$ and $\omega$ denote respectively the Möbius 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 $\sum_{P^-(n)\in \scr P}μ(n)ω(n)/n=0$ 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.