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arXiv · 2409.02754

On a family of arithmetic series related to the Möbius function

Gérald Tenenbaum
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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 $\sum_{P^-(n)\in \scr P}μ(n)ω(n)/n=0$ 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.

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Gérald Tenenbaum. 2026-03-03. On a family of arithmetic series related to the Möbius function. https://arxiv.org/abs/2409.02754

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