On a family of arithmetic series related to the Möbius function
Résumé
Let P^-(n) denote the smallest prime factor of a natural integer n>1. Furthermore let µ and ω denote respectively the Möbius function and the number of distinct prime factors function. We show that, given any set P of prime numbers with a natural density, we have \sum_{P-(n)∈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 P is an arithmetic progression.
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