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On a series related to the Möbius function

  • Deep Thakur
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Abstract

For a natural number n>1 let P-(n) denote its smallest prime divisor. Alladi and Johnson proved that ∑P−(n)≡ l Modk μ(n)ω(n)/n = 0 for (l,k)=1, and Tenenbaum showed that the partial sums of this series up to X are Oε((loglog X)5/3+ε/log X). In this article, we prove that these partial sums are in fact equal to C(k,l)/log X + O((loglog X)4/(log X)2), with an explicit constant C(k,l). We also investigate the behaviour of these constants. In particular, we show under GRH that the number of reduced classes lModk for which C(k,l)<0 is $φ(k)(1- o(1))$ as k→∞.

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Cite

BibTeXRIS
BibTeX
@misc{thakur2026series,
  title = {{On a series related to the Möbius function}},
  author = {Deep Thakur},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23660},
  eprint = {2610.23660},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23660},
}
RIS
TY  - UNPB
TI  - On a series related to the Möbius function
AU  - Deep Thakur
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23660
UR  - https://arxiv.org/abs/2610.23660
AB  - For a natural number n>1 let P^-(n) denote its smallest prime divisor. Alladi and Johnson proved that ∑_P^−(n)≡ l Modk μ(n)ω(n)/n = 0 for (l,k)=1, and Tenenbaum showed that the partial sums of this series up to X are O_ε((loglog X)^5/3+ε/log X). In this article, we prove that these partial sums are in fact equal to C(k,l)/log X + O((loglog X)^4/(log X)^2), with an explicit constant C(k,l). We also investigate the behaviour of these constants. In particular, we show under GRH that the number of reduced classes lModk for which C(k,l)<0 is $φ(k)(1- o(1))$ as k→∞.
ER  -