---
OA_place: repository
OA_type: green
_id: '22193'
abstract:
- lang: eng
  text: "Gross and Smith have put forward generalizations of Hardy–Littlewood twin
    prime\r\nconjectures for algebraic number fields. We estimate the behaviour of
    sums of a singular series that arises in these conjectures, up to lower-order
    terms. More exactly,\r\nwhere S(η) is the singular series, we find asymptotic
    formulas for smoothed sums of\r\nS(η)−1. Based upon Gross and Smith’s conjectures,
    we use our result to suggest that\r\nfor large enough ‘short intervals’ in an
    algebraic number field K, the variance of counts\r\nof prime elements in a random
    short interval deviates from a Cramér model prediction\r\nby a universal factor,
    independent of K. The conjecture over number fields generalizes\r\na classical
    conjecture of Goldston and Montgomery over the integers. Numerical data\r\nare
    provided supporting the conjecture."
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
- first_name: Brad
  full_name: Rodgers, Brad
  last_name: Rodgers
- first_name: Edva
  full_name: Roditty-Gershon, Edva
  last_name: Roditty-Gershon
citation:
  ama: Kuperberg VZ, Rodgers B, Roditty-Gershon E. Sums of singular series and primes
    in short intervals in algebraic number fields. <i>The Ramanujan Journal</i>. 2022;58(2):291-317.
    doi:<a href="https://doi.org/10.1007/s11139-022-00561-9">10.1007/s11139-022-00561-9</a>
  apa: Kuperberg, V. Z., Rodgers, B., &#38; Roditty-Gershon, E. (2022). Sums of singular
    series and primes in short intervals in algebraic number fields. <i>The Ramanujan
    Journal</i>. Springer Nature. <a href="https://doi.org/10.1007/s11139-022-00561-9">https://doi.org/10.1007/s11139-022-00561-9</a>
  chicago: Kuperberg, Vivian Zieve, Brad Rodgers, and Edva Roditty-Gershon. “Sums
    of Singular Series and Primes in Short Intervals in Algebraic Number Fields.”
    <i>The Ramanujan Journal</i>. Springer Nature, 2022. <a href="https://doi.org/10.1007/s11139-022-00561-9">https://doi.org/10.1007/s11139-022-00561-9</a>.
  ieee: V. Z. Kuperberg, B. Rodgers, and E. Roditty-Gershon, “Sums of singular series
    and primes in short intervals in algebraic number fields,” <i>The Ramanujan Journal</i>,
    vol. 58, no. 2. Springer Nature, pp. 291–317, 2022.
  ista: Kuperberg VZ, Rodgers B, Roditty-Gershon E. 2022. Sums of singular series
    and primes in short intervals in algebraic number fields. The Ramanujan Journal.
    58(2), 291–317.
  mla: Kuperberg, Vivian Zieve, et al. “Sums of Singular Series and Primes in Short
    Intervals in Algebraic Number Fields.” <i>The Ramanujan Journal</i>, vol. 58,
    no. 2, Springer Nature, 2022, pp. 291–317, doi:<a href="https://doi.org/10.1007/s11139-022-00561-9">10.1007/s11139-022-00561-9</a>.
  short: V.Z. Kuperberg, B. Rodgers, E. Roditty-Gershon, The Ramanujan Journal 58
    (2022) 291–317.
date_created: 2026-06-29T12:57:02Z
date_published: 2022-06-01T00:00:00Z
date_updated: 2026-07-14T10:59:34Z
day: '01'
doi: 10.1007/s11139-022-00561-9
extern: '1'
external_id:
  arxiv:
  - '2001.09513'
intvolume: '        58'
issue: '2'
keyword:
- Primes
- Short intervals
- Ramanujan sums
- Singular series
- Number fields
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2001.09513
mathsc:
- 11N05
- 11R47
month: '06'
oa: 1
oa_version: Preprint
page: 291-317
publication: The Ramanujan Journal
publication_identifier:
  eissn:
  - 1572-9303
  issn:
  - 1382-4090
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
status: public
title: Sums of singular series and primes in short intervals in algebraic number fields
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 58
year: '2022'
...