---
OA_place: repository
OA_type: green
_id: '22191'
abstract:
- lang: eng
  text: "Montgomery and Soundararajan showed that the distribution of ψ(x+H)−ψ(x),
    for 0≤ x ≤ N, is approximately normal with mean ∼ H and variance ∼ H log(N/H),
    when N\r\nδ ≤ H ≤ N\r\n1−δ\r\n. Their work depends\r\non showing that sums Rk
    (h) of k-term singular series are µk (−h log h + Ah)\r\nk/2 + Ok (h\r\nk/2−1/(7k)+ε\r\n),\r\nwhere
    A is a constant and µk are the Gaussian moment constants. We study lower-order
    terms in the size\r\nof these moments. We conjecture that when k is odd, Rk (h)
    ≍ h\r\n(k−1)/2\r\n(log h)\r\n(k+1)/2\r\n. We prove an upper\r\nbound with the
    correct power of h when k = 3, and prove analogous upper bounds in the function
    field\r\nsetting when k = 3 and k = 5. We provide further evidence for this conjecture
    in the form of numerical\r\ncomputations."
article_processing_charge: No
article_type: original
author:
- first_name: Vivian Zieve
  full_name: Kuperberg, Vivian Zieve
  id: c3bac823-112d-11f0-a3f5-c264f852e697
  last_name: Kuperberg
citation:
  ama: Kuperberg VZ. Odd moments in the distribution of primes. <i>Algebra &#38; Number
    Theory</i>. 2025;19(4):617-666. doi:<a href="https://doi.org/10.2140/ant.2025.19.617">10.2140/ant.2025.19.617</a>
  apa: Kuperberg, V. Z. (2025). Odd moments in the distribution of primes. <i>Algebra
    &#38; Number Theory</i>. Mathematical Sciences Publishers. <a href="https://doi.org/10.2140/ant.2025.19.617">https://doi.org/10.2140/ant.2025.19.617</a>
  chicago: Kuperberg, Vivian Zieve. “Odd Moments in the Distribution of Primes.” <i>Algebra
    &#38; Number Theory</i>. Mathematical Sciences Publishers, 2025. <a href="https://doi.org/10.2140/ant.2025.19.617">https://doi.org/10.2140/ant.2025.19.617</a>.
  ieee: V. Z. Kuperberg, “Odd moments in the distribution of primes,” <i>Algebra &#38;
    Number Theory</i>, vol. 19, no. 4. Mathematical Sciences Publishers, pp. 617–666,
    2025.
  ista: Kuperberg VZ. 2025. Odd moments in the distribution of primes. Algebra &#38;
    Number Theory. 19(4), 617–666.
  mla: Kuperberg, Vivian Zieve. “Odd Moments in the Distribution of Primes.” <i>Algebra
    &#38; Number Theory</i>, vol. 19, no. 4, Mathematical Sciences Publishers, 2025,
    pp. 617–66, doi:<a href="https://doi.org/10.2140/ant.2025.19.617">10.2140/ant.2025.19.617</a>.
  short: V.Z. Kuperberg, Algebra &#38; Number Theory 19 (2025) 617–666.
date_created: 2026-06-29T12:56:09Z
date_published: 2025-03-24T00:00:00Z
date_updated: 2026-07-14T10:52:02Z
day: '24'
doi: 10.2140/ant.2025.19.617
extern: '1'
external_id:
  unknown:
  - '2109.03767'
fulldoi: https://doi.org/10.2140/ant.2025.19.617
intvolume: '        19'
issue: '4'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2109.03767
mathsc:
- 11N05
- 11N13
month: '03'
oa: 1
oa_version: Preprint
page: 617-666
publication: Algebra & Number Theory
publication_identifier:
  eissn:
  - 1944-7833
  issn:
  - 1937-0652
publication_status: published
publisher: Mathematical Sciences Publishers
quality_controlled: '1'
scopus_import: '1'
status: public
title: Odd moments in the distribution of primes
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 19
year: '2025'
...