On a family of arithmetic series related to the Möbius function

2024/09/04 by Tenenbaum, Gérald
#11N 37 #11N25 #11N56 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2409.02754

Abstract

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 \scr P of prime numbers with a natural density, we have ∑_{P^-(n)∈ \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.

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