---
OA_place: repository
OA_type: green
_id: '22192'
abstract:
- lang: eng
  text: "In 1976, Gallagher showed that the Hardy–Littlewood conjectures on prime
    k-tuples imply that the\r\ndistribution of primes in log-size intervals is Poissonian.
    He did so by computing average values\r\nof the singular series constants over
    different sets of a fixed size k contained in an interval [1,h]\r\nas h → ∞, and
    then using this average to compute moments of the distribution of primes. In this\r\npaper,
    we study averages where k is relatively large with respect to h. We then apply
    these averages\r\nto the tail of the distribution. For example, we show, assuming
    appropriate Hardy–Littlewood\r\nconjectures and in certain ranges of the parameters,
    the number of intervals [n,n + λlogx] with\r\nn ≤ x containing at least k primes
    is ≪ x exp(−k/(λe))."
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Vivian Zieve
  full_name: Kuperberg, Vivian Zieve
  id: c3bac823-112d-11f0-a3f5-c264f852e697
  last_name: Kuperberg
citation:
  ama: Kuperberg VZ. Sums of singular series with large sets and the tail of the distribution
    of primes. <i>The Quarterly Journal of Mathematics</i>. 2023;74(4):1457-1479.
    doi:<a href="https://doi.org/10.1093/qmath/haad030">10.1093/qmath/haad030</a>
  apa: Kuperberg, V. Z. (2023). Sums of singular series with large sets and the tail
    of the distribution of primes. <i>The Quarterly Journal of Mathematics</i>. Oxford
    University Press. <a href="https://doi.org/10.1093/qmath/haad030">https://doi.org/10.1093/qmath/haad030</a>
  chicago: Kuperberg, Vivian Zieve. “Sums of Singular Series with Large Sets and the
    Tail of the Distribution of Primes.” <i>The Quarterly Journal of Mathematics</i>.
    Oxford University Press, 2023. <a href="https://doi.org/10.1093/qmath/haad030">https://doi.org/10.1093/qmath/haad030</a>.
  ieee: V. Z. Kuperberg, “Sums of singular series with large sets and the tail of the
    distribution of primes,” <i>The Quarterly Journal of Mathematics</i>, vol. 74,
    no. 4. Oxford University Press, pp. 1457–1479, 2023.
  ista: Kuperberg VZ. 2023. Sums of singular series with large sets and the tail of the
    distribution of primes. The Quarterly Journal of Mathematics. 74(4), 1457–1479.
  mla: Kuperberg, Vivian Zieve. “Sums of Singular Series with Large Sets and the Tail
    of the Distribution of Primes.” <i>The Quarterly Journal of Mathematics</i>, vol.
    74, no. 4, Oxford University Press, 2023, pp. 1457–79, doi:<a href="https://doi.org/10.1093/qmath/haad030">10.1093/qmath/haad030</a>.
  short: V.Z. Kuperberg, The Quarterly Journal of Mathematics 74 (2023) 1457–1479.
date_created: 2026-06-29T12:56:30Z
date_published: 2023-12-01T00:00:00Z
date_updated: 2026-07-14T10:55:52Z
day: '01'
doi: 10.1093/qmath/haad030
extern: '1'
external_id:
  arxiv:
  - '2210.09775'
intvolume: '        74'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2210.09775
month: '12'
oa: 1
oa_version: Preprint
page: 1457-1479
publication: The Quarterly Journal of Mathematics
publication_identifier:
  eissn:
  - 1464-3847
  issn:
  - 0033-5606
publication_status: published
publisher: Oxford University Press
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sums of singular series with large sets and the tail of the distribution of primes
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 74
year: '2023'
...