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On a family of arithmetic series related to the Möbius function

Gérald Tenenbaum

TL;DR

This work extends Alladi and Johnson's result by showing that for any set of primes with natural density, the sum over numbers whose smallest prime factor lies in that set satisfies $\sum_{P^-(n)\in{\mathcal P}}{\mu(n)\omega(n)\over n}=0$, and provides an effective rate of convergence. The authors combine Möbius inversion, Dirichlet series, Perron's formula, and Selberg-Delange analysis, decomposing contributions into primes up to a threshold and those beyond, with the Dickman function governing main-term behavior. A key outcome is a quantitative bound $A^-(x,y)+A^+(x,y) \ll \varepsilon_{\mathcal P}^*(y)\log u+1/u$, where $u=(\log x)/\log y$, yielding zero-sum convergence under broad density assumptions and offering explicit error control. Special cases for ${\mathcal P}$ equal to all primes or a singleton illustrate asymptotics and highlight limitations of naively reconstructing global behavior from restricted prime subsets. The results generalize the underlying small-vs-large prime duality and sharpen our understanding of how prime-factor structure interacts with prescribed prime-subsets in arithmetic sums.

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.

On a family of arithmetic series related to the Möbius function

TL;DR

This work extends Alladi and Johnson's result by showing that for any set of primes with natural density, the sum over numbers whose smallest prime factor lies in that set satisfies , and provides an effective rate of convergence. The authors combine Möbius inversion, Dirichlet series, Perron's formula, and Selberg-Delange analysis, decomposing contributions into primes up to a threshold and those beyond, with the Dickman function governing main-term behavior. A key outcome is a quantitative bound , where , yielding zero-sum convergence under broad density assumptions and offering explicit error control. Special cases for equal to all primes or a singleton illustrate asymptotics and highlight limitations of naively reconstructing global behavior from restricted prime subsets. The results generalize the underlying small-vs-large prime duality and sharpen our understanding of how prime-factor structure interacts with prescribed prime-subsets in arithmetic sums.

Abstract

Let denote the smallest prime factor of a natural integer . Furthermore let and denote respectively the Möbius function and the number of distinct prime factors function. We show that, given any set of prime numbers with a natural density, we have and provide a effective estimate for the rate of convergence. This extends a recent result of Alladi and Johnson, who considered the case when is an arithmetic progression.
Paper Structure (3 sections, 6 theorems, 69 equations)

This paper contains 3 sections, 6 theorems, 69 equations.

Table of Contents

  1. Introduction and statements
  2. Proof of Theorem \ref{['thgen']}
  3. Special cases

Key Result

Theorem 1.1

Let ${\EuScript P}$ be a set of prime numbers satisfying, for suitable $\delta\in[0,1]$, Then Moreover, for any fixed $b>5/3$ and uniformly for ${\rm e}^{(\log_2x)^b}\leqslant y\leqslant \sqrt{x}$, we have where $\varepsilon_{\EuScript P}^*(y):=\sup_{t>y}|\varepsilon_{\EuScript P}(t)|$.

Theorems & Definitions (11)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 1 more