---
OA_place: repository
OA_type: green
_id: '22195'
abstract:
- lang: eng
  text: 'Sums of the singular series constants that appear in the Hardy–Littlewood
    k-tuples conjectures have long been studied in connection to the distribution
    of primes. We study constrained sums of singular series, where the sum is taken
    over sets whose elements are specified modulo r or weighted by smooth functions.
    We show that the value of the sum is governed by incidences modulo r of elements
    of the set in the case of arithmetic progressions and by pairings of the smooth
    functions in the case of weights. These sums shed light on sums of singular series
    in other formats. '
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 along arithmetic progressions and with
    smooth weights. <i>International Journal of Number Theory</i>. 2025;21(01):53-74.
    doi:<a href="https://doi.org/10.1142/s1793042125500046">10.1142/s1793042125500046</a>
  apa: Kuperberg, V. Z. (2025). Sums of singular series along arithmetic progressions
    and with smooth weights. <i>International Journal of Number Theory</i>. World
    Scientific Publishing. <a href="https://doi.org/10.1142/s1793042125500046">https://doi.org/10.1142/s1793042125500046</a>
  chicago: Kuperberg, Vivian Zieve. “Sums of Singular Series along Arithmetic Progressions
    and with Smooth Weights.” <i>International Journal of Number Theory</i>. World
    Scientific Publishing, 2025. <a href="https://doi.org/10.1142/s1793042125500046">https://doi.org/10.1142/s1793042125500046</a>.
  ieee: V. Z. Kuperberg, “Sums of singular series along arithmetic progressions and
    with smooth weights,” <i>International Journal of Number Theory</i>, vol. 21,
    no. 01. World Scientific Publishing, pp. 53–74, 2025.
  ista: Kuperberg VZ. 2025. Sums of singular series along arithmetic progressions
    and with smooth weights. International Journal of Number Theory. 21(01), 53–74.
  mla: Kuperberg, Vivian Zieve. “Sums of Singular Series along Arithmetic Progressions
    and with Smooth Weights.” <i>International Journal of Number Theory</i>, vol.
    21, no. 01, World Scientific Publishing, 2025, pp. 53–74, doi:<a href="https://doi.org/10.1142/s1793042125500046">10.1142/s1793042125500046</a>.
  short: V.Z. Kuperberg, International Journal of Number Theory 21 (2025) 53–74.
date_created: 2026-06-29T12:57:46Z
date_published: 2025-01-01T00:00:00Z
date_updated: 2026-07-14T11:04:58Z
day: '01'
doi: 10.1142/s1793042125500046
extern: '1'
external_id:
  arxiv:
  - '2301.06095'
intvolume: '        21'
issue: '01'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2301.06095
month: '01'
oa: 1
oa_version: Preprint
page: 53-74
publication: International Journal of Number Theory
publication_identifier:
  eissn:
  - 1793-7310
  issn:
  - 1793-0421
publication_status: published
publisher: World Scientific Publishing
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sums of singular series along arithmetic progressions and with smooth weights
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 21
year: '2025'
...