---
OA_place: publisher
OA_type: hybrid
PlanS_conform: '1'
_id: '22644'
abstract:
- lang: eng
  text: "We analyze the average behavior of various arithmetic functions at the values
    of degree \U0001D451 binary forms ordered by height, with probability 1. This
    approach yields averaged versions of the Chowla conjecture and the Bateman–Horn
    conjecture for random binary forms. Furthermore, we show that the rational Hasse
    principle holds for almost all Châtelet varieties defined by a fixed norm form
    of degree \U0001D452 and by varying binary forms of fixed degree \U0001D451, provided
    \U0001D452 divides \U0001D451. This proves an average version of a conjecture
    of Colliot-Thélène."
acknowledgement: I am deeply grateful to my advisor Tim Browning for suggesting this
  problem and for the many valuable discussions that shaped this work. I would also
  like to thank Efthymios Sofos, Matteo Verzobio, and Shuntaro Yamagishi for discussions
  and insights that contributed to this paper. I am also very grateful to the anonymous
  referee for their careful reading and for the considerable effort they put into
  improving the manuscript.
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Yijie
  full_name: Diao, Yijie
  id: 7b7eb4ca-eb2c-11ec-b98b-accec0b20c3b
  last_name: Diao
  orcid: 0000-0002-4989-5330
citation:
  ama: Diao Y. Liouville function, von Mangoldt function, and norm forms at random
    binary forms. <i>Glasgow Mathematical Journal</i>. 2026:1-34. doi:<a href="https://doi.org/10.1017/s0017089526101074">10.1017/s0017089526101074</a>
  apa: Diao, Y. (2026). Liouville function, von Mangoldt function, and norm forms
    at random binary forms. <i>Glasgow Mathematical Journal</i>. Cambridge University
    Press. <a href="https://doi.org/10.1017/s0017089526101074">https://doi.org/10.1017/s0017089526101074</a>
  chicago: Diao, Yijie. “Liouville Function, von Mangoldt Function, and Norm Forms
    at Random Binary Forms.” <i>Glasgow Mathematical Journal</i>. Cambridge University
    Press, 2026. <a href="https://doi.org/10.1017/s0017089526101074">https://doi.org/10.1017/s0017089526101074</a>.
  ieee: Y. Diao, “Liouville function, von Mangoldt function, and norm forms at random
    binary forms,” <i>Glasgow Mathematical Journal</i>. Cambridge University Press,
    pp. 1–34, 2026.
  ista: Diao Y. 2026. Liouville function, von Mangoldt function, and norm forms at
    random binary forms. Glasgow Mathematical Journal., 1–34.
  mla: Diao, Yijie. “Liouville Function, von Mangoldt Function, and Norm Forms at
    Random Binary Forms.” <i>Glasgow Mathematical Journal</i>, Cambridge University
    Press, 2026, pp. 1–34, doi:<a href="https://doi.org/10.1017/s0017089526101074">10.1017/s0017089526101074</a>.
  short: Y. Diao, Glasgow Mathematical Journal (2026) 1–34.
corr_author: '1'
das_tickbox: '0'
date_created: 2026-08-04T06:15:04Z
date_published: 2026-07-21T00:00:00Z
date_updated: 2026-08-04T06:28:01Z
day: '21'
ddc:
- '500'
department:
- _id: TiBr
- _id: GradSch
doi: 10.1017/s0017089526101074
external_id:
  arxiv:
  - '2506.18065'
has_accepted_license: '1'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1017/S0017089526101074
mathsc:
- 11N32
- 11N37
- 11D57
- 11G35
month: '07'
oa: 1
oa_version: Published Version
page: 1-34
publication: Glasgow Mathematical Journal
publication_identifier:
  eissn:
  - 1469-509X
  issn:
  - 0017-0895
publication_status: epub_ahead
publisher: Cambridge University Press
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Liouville function, von Mangoldt function, and norm forms at random binary
  forms
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2026'
...