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PreprintCC BY 4.0arXiv

On the hyperbolic prime number theorem

  • Alisa Sedunova
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Abstract

Friedlander and Iwaniec proved that the number of points of the orbit {γi : γ∈SL2(Z)} lying at a distance p−2 from the origin i of the upper half-plane, with p≤ x prime, is of order x/log x; the upper bound is unconditional, while the lower one rests on a strong hypothesis concerning the distribution of primes in arithmetic progressions. We consider instead square-free distances and prove unconditionally an asymptotic formula, whose main term is of order x. Employing the weighted linear sieve, we also show unconditionally that the square-free distances n with at most $7$ prime factors contribute ≫ x/log x. Finally, assuming that the sequence r(n−2)r(n+2) has level of distribution xθ in arithmetic progressions, where r(n) is the number of ways to write n as a sum of two squares, we obtain the same lower bound for the distances with at most N prime factors (the value $7$ corresponds to θ=1/6).

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Cite

BibTeXRIS
BibTeX
@misc{sedunova2026hyperbolic,
  title = {{On the hyperbolic prime number theorem}},
  author = {Alisa Sedunova},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.28200},
  eprint = {2609.28200},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.28200},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On the hyperbolic prime number theorem
AU  - Alisa Sedunova
PY  - 2026
DA  - 2026/09/23
PB  - arXiv
DO  - 10.48550/arxiv.2609.28200
UR  - https://arxiv.org/abs/2609.28200
AB  - Friedlander and Iwaniec proved that the number of points of the orbit {γi : γ∈SL_2(Z)} lying at a distance p−2 from the origin i of the upper half-plane, with p≤ x prime, is of order x/log x; the upper bound is unconditional, while the lower one rests on a strong hypothesis concerning the distribution of primes in arithmetic progressions. We consider instead square-free distances and prove unconditionally an asymptotic formula, whose main term is of order x. Employing the weighted linear sieve, we also show unconditionally that the square-free distances n with at most $7$ prime factors contribute ≫ x/log x. Finally, assuming that the sequence r(n−2)r(n+2) has level of distribution x^θ in arithmetic progressions, where r(n) is the number of ways to write n as a sum of two squares, we obtain the same lower bound for the distances with at most N prime factors (the value $7$ corresponds to θ=1/6).
ER  -