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arXiv · 2409.02754

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

Gérald Tenenbaum
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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.

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Gérald Tenenbaum. 2024-09-04. On a family of arithmetic series related to the M\"obius function. https://arxiv.org/abs/2409.02754

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