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PreprintCC BY 4.0arXiv

On Landau's "local method" for obtaining zero-free regions

  • Fredrik Broucke
  • Szilárd Gy. Révész
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Abstract

We extend Landau's local analytic method for the deduction of zero-free region to a very general setting, and as a particular consequence, obtain a zero-free region and corresponding Prime Number Theorem for Beurling number systems with well-behaved integers. We also discuss the sharpness of the method and prove estimates for the local number of zeros, as well as a clustering result, which roughly put says that zeros close to the 1-line cannot be isolated but must occur in clusters.

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Cite

BibTeXRIS
BibTeX
@misc{broucke2026landaus,
  title = {{On Landau's "local method" for obtaining zero-free regions}},
  author = {Fredrik Broucke and Szilárd Gy. Révész},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03356},
  eprint = {2611.03356},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03356},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On Landau's "local method" for obtaining zero-free regions
AU  - Fredrik Broucke
AU  - Szilárd Gy. Révész
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03356
UR  - https://arxiv.org/abs/2611.03356
AB  - We extend Landau's local analytic method for the deduction of zero-free region to a very general setting, and as a particular consequence, obtain a zero-free region and corresponding Prime Number Theorem for Beurling number systems with well-behaved integers. We also discuss the sharpness of the method and prove estimates for the local number of zeros, as well as a clustering result, which roughly put says that zeros close to the 1-line cannot be isolated but must occur in clusters.
ER  -