---
OA_place: repository
OA_type: green
_id: '22203'
abstract:
- lang: eng
  text: "We prove near-optimal upper bounds for the oddmoments of the distribution
    of coprime residues inshort intervals, confirming a conjecture of Montgomeryand
    Vaughan. As an application, we prove near-optimalupper bounds for the average
    of the refined singularseries in the Hardy–Littlewood conjectures concerningthe
    number of prime \U0001D458-tuples for \U0001D458 odd. The mainnew ingredient is
    a near-optimal upper bound for thenumber of solutions to ∑1 ⩽\U0001D456 ⩽\U0001D458\U0001D44E
    \U0001D456\U0001D45E\U0001D456∈ ℤ when \U0001D458 is odd,with gcd(\U0001D44E\U0001D456
    , \U0001D45E\U0001D456 ) = 1 and restrictions on the size of thenumerators and
    denominators, which is of indepen-dent interest."
article_number: e70068
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Thomas F.
  full_name: Bloom, Thomas F.
  last_name: Bloom
- first_name: Vivian Zieve
  full_name: Kuperberg, Vivian Zieve
  id: c3bac823-112d-11f0-a3f5-c264f852e697
  last_name: Kuperberg
citation:
  ama: Bloom TF, Kuperberg VZ. Odd moments and adding fractions. <i>Proceedings of
    the London Mathematical Society</i>. 2025;131(1). doi:<a href="https://doi.org/10.1112/plms.70068">10.1112/plms.70068</a>
  apa: Bloom, T. F., &#38; Kuperberg, V. Z. (2025). Odd moments and adding fractions.
    <i>Proceedings of the London Mathematical Society</i>. Wiley. <a href="https://doi.org/10.1112/plms.70068">https://doi.org/10.1112/plms.70068</a>
  chicago: Bloom, Thomas F., and Vivian Zieve Kuperberg. “Odd Moments and Adding Fractions.”
    <i>Proceedings of the London Mathematical Society</i>. Wiley, 2025. <a href="https://doi.org/10.1112/plms.70068">https://doi.org/10.1112/plms.70068</a>.
  ieee: T. F. Bloom and V. Z. Kuperberg, “Odd moments and adding fractions,” <i>Proceedings
    of the London Mathematical Society</i>, vol. 131, no. 1. Wiley, 2025.
  ista: Bloom TF, Kuperberg VZ. 2025. Odd moments and adding fractions. Proceedings
    of the London Mathematical Society. 131(1), e70068.
  mla: Bloom, Thomas F., and Vivian Zieve Kuperberg. “Odd Moments and Adding Fractions.”
    <i>Proceedings of the London Mathematical Society</i>, vol. 131, no. 1, e70068,
    Wiley, 2025, doi:<a href="https://doi.org/10.1112/plms.70068">10.1112/plms.70068</a>.
  short: T.F. Bloom, V.Z. Kuperberg, Proceedings of the London Mathematical Society
    131 (2025).
date_created: 2026-06-29T13:00:46Z
date_published: 2025-07-01T00:00:00Z
date_updated: 2026-07-14T11:45:18Z
day: '01'
doi: 10.1112/plms.70068
extern: '1'
external_id:
  arxiv:
  - '2312.09021'
intvolume: '       131'
issue: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2312.09021
mathsc:
- 11N69
- 11N05
- 14G05
month: '07'
oa: 1
oa_version: Preprint
publication: Proceedings of the London Mathematical Society
publication_identifier:
  eissn:
  - 1460-244X
  issn:
  - 0024-6115
publication_status: published
publisher: Wiley
quality_controlled: '1'
scopus_import: '1'
status: public
title: Odd moments and adding fractions
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 131
year: '2025'
...