<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$ P^-(n)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>P</mi> <mo>-</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the smallest prime factor of a natural integer <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore let <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mu $</EquationSource> <EquationSource Format="MATHML"><math> <mi>μ</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\omega $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ω</mi> </math></EquationSource> </InlineEquation> denote respectively the Möbius function and the number of distinct prime factors function. We show that, given any set <InlineEquation ID="IEq5"> <EquationSource Format="TEX">${\mathcal {P}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">P</mi> </math></EquationSource> </InlineEquation> of prime numbers with a natural density, we have <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\sum _{P^-(n)\in {\mathcal {P}}}\mu (n)\omega (n)/n=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>∑</mo> <mrow> <msup> <mi>P</mi> <mo>-</mo> </msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <mi mathvariant="script">P</mi> </mrow> </msub> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mi>ω</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">/</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and provide a effective estimate for the rate of convergence. This extends a recent result of Alladi and Johnson, who considered the case when <InlineEquation ID="IEq7"> <EquationSource Format="TEX">${\mathcal {P}}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">P</mi> </math></EquationSource> </InlineEquation> is an arithmetic progression.</p>

On a family of arithmetic series related to the Möbius function

期刊论文
发表日期：

2026年3月17日

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Gérald Tenenbaum

摘要

Let $ P^-(n)$ $P^{-} (n)$ denote the smallest prime factor of a natural integer $n>1$ $n > 1$ . Furthermore let $\mu $ $μ$ and $\omega $ $ω$ denote respectively the Möbius function and the number of distinct prime factors function. We show that, given any set ${\mathcal {P}}$ $P$ of prime numbers with a natural density, we have $\sum _{P^-(n)\in {\mathcal {P}}}\mu (n)\omega (n)/n=0$ $\sum_{P^{-} (n) \in P} μ (n) ω (n) / n = 0$ and provide a effective estimate for the rate of convergence. This extends a recent result of Alladi and Johnson, who considered the case when ${\mathcal {P}}$ $P$ is an arithmetic progression.