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PreprintarXiv licencearXiv

Improved average and almost-all bounds for G(n)

  • Chiara Bellotti
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Abstract

In this paper we study the least positive integer G(n) such that the integers a≤ G(n) with (a,n)=1 generate (ℤ/nℤ)×. We prove that, for every ε>0, ∑n≤ xG(n)≪ε x(log x)8/3+ε, improving the previously known bound ≪ x(log x)97. We also prove that G(n)≤(log n)2 for almost all n, unconditionally. Thus, for almost all n, we obtain the same logarithmic exponent $2$ as in the classical pointwise bound under GRH.

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Cite

BibTeXRIS
BibTeX
@misc{bellotti2026improved,
  title = {{Improved average and almost-all bounds for $G(n)$}},
  author = {Chiara Bellotti},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10177},
  eprint = {2610.10177},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10177},
}
RIS
TY  - UNPB
TI  - Improved average and almost-all bounds for G(n)
AU  - Chiara Bellotti
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10177
UR  - https://arxiv.org/abs/2610.10177
AB  - In this paper we study the least positive integer G(n) such that the integers a≤ G(n) with (a,n)=1 generate (ℤ/nℤ)^×. We prove that, for every ε>0, ∑_n≤ xG(n)≪_ε x(log x)^8/3+ε, improving the previously known bound ≪ x(log x)^97. We also prove that G(n)≤(log n)^2 for almost all n, unconditionally. Thus, for almost all n, we obtain the same logarithmic exponent $2$ as in the classical pointwise bound under GRH.
ER  -