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Peer-reviewedCC BYMathematics of Computation

Möbius function and primes: an identity factory with applications

  • Olivier Ramaré
  • Sebastian Zuniga Alterman
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Abstract

We investigate the sums ∑ n ≤ X , ( n , q ) = 1 μ ( n ) n s log k ⁡ ( X n ) \sum _{n\le X, (n,q)=1}\frac {\mu (n)}{n^{s}}\log ^{k}\left (\frac {X}{n}\right ) , where k ∈ { 0 , 1 } k\in \{0,1\} , s ∈ C s\in \mathbb {C} , ℜ s > 0 \Re s>0 . Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this, we develop a broad framework of identities that we use to derive several applications. Building on similar principles, we also provide an appendix establishing the inequality ∑ n ≤ X Λ ( n ) / n ≤ log ⁡ X \sum _{n\le X}\Lambda (n)/n\le \log X , valid for any X ≥ 1 X\geq 1 .

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Cite

BibTeXRIS
BibTeX
@article{ramare2026möbius,
  title = {{Möbius function and primes: an identity factory with applications}},
  author = {Olivier Ramaré and Sebastian Zuniga Alterman},
  journal = {Mathematics of Computation},
  year = {2026},
  month = jul,
  doi = {10.1090/mcom/4216},
  url = {https://doi.org/10.1090/mcom/4216},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Möbius function and primes: an identity factory with applications
AU  - Olivier Ramaré
AU  - Sebastian Zuniga Alterman
PY  - 2026
DA  - 2026/07/23
JO  - Mathematics of Computation
DO  - 10.1090/mcom/4216
UR  - https://doi.org/10.1090/mcom/4216
AB  - We investigate the sums ∑ n ≤ X , ( n , q ) = 1 μ ( n ) n s log k ⁡ ( X n ) \sum _{n\le X, (n,q)=1}\frac {\mu (n)}{n^{s}}\log ^{k}\left (\frac {X}{n}\right ) , where k ∈ { 0 , 1 } k\in \{0,1\} , s ∈ C s\in \mathbb {C} , ℜ s > 0 \Re s>0 . Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this, we develop a broad framework of identities that we use to derive several applications. Building on similar principles, we also provide an appendix establishing the inequality ∑ n ≤ X Λ ( n ) / n ≤ log ⁡ X \sum _{n\le X}\Lambda (n)/n\le \log X , valid for any X ≥ 1 X\geq 1 .
SN  - 0025-5718
SN  - 1088-6842
ER  -