---
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'
fulldoi: https://doi.org/10.1017/s0017089526101074
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'
...
---
OA_place: publisher
OA_type: hybrid
_id: '20850'
abstract:
- lang: eng
  text: We provide an estimate for the number of nontrivial integer points on the
    Pellian surface t^2 - du^2 = 1 in a bounded region. We give a lower bound on the
    size of fundamental solutions for almost all d in a certain class, based on a
    recent conjecture of Browning and Wilsch about integer points on log K3 surfaces.
    We also obtain an upper bound on the average of class number in this class, assuming
    the same conjecture.
- lang: fre
  text: "Nous donnons une estimation du nombre de points entiers non triviaux sur
    la surface pellienne \r\nt^2 - du^2 = 1 dans une région bornée. Nous établissons
    une borne inférieure pour la taille des solutions fondamentales pour presque tout
    d appartenant à une certaine classe, en nous fondant sur une conjecture récente
    de Browning et Wilsch concernant les points entiers sur les surfaces log K3. Nous
    obtenons également une borne supérieure pour la moyenne du nombre de classes dans
    cette classe, sous la même hypothèse conjecturale."
acknowledgement: The author would like to thank his supervisor Tim Browning for suggesting
  this project and many helpful conversations and useful comments. Moreover, he is
  grateful to Jakob Glas, Damaris Schindler, Igor Shparlinski, Matteo Verzobio, Victor
  Wang, Florian Wilsch and Shuntaro Yamagishi for taking their time to answer his
  questions and their valuable suggestions.
article_processing_charge: Yes (in subscription journal)
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. Class numbers and integer points on some Pellian surfaces. <i>Journal
    de theorie des nombres de Bordeaux</i>. 2025;37(3):973-988. doi:<a href="https://doi.org/10.5802/jtnb.1348">10.5802/jtnb.1348</a>
  apa: Diao, Y. (2025). Class numbers and integer points on some Pellian surfaces.
    <i>Journal de Theorie Des Nombres de Bordeaux</i>. Université de Bordeaux. <a
    href="https://doi.org/10.5802/jtnb.1348">https://doi.org/10.5802/jtnb.1348</a>
  chicago: Diao, Yijie. “Class Numbers and Integer Points on Some Pellian Surfaces.”
    <i>Journal de Theorie Des Nombres de Bordeaux</i>. Université de Bordeaux, 2025.
    <a href="https://doi.org/10.5802/jtnb.1348">https://doi.org/10.5802/jtnb.1348</a>.
  ieee: Y. Diao, “Class numbers and integer points on some Pellian surfaces,” <i>Journal
    de theorie des nombres de Bordeaux</i>, vol. 37, no. 3. Université de Bordeaux,
    pp. 973–988, 2025.
  ista: Diao Y. 2025. Class numbers and integer points on some Pellian surfaces. Journal
    de theorie des nombres de Bordeaux. 37(3), 973–988.
  mla: Diao, Yijie. “Class Numbers and Integer Points on Some Pellian Surfaces.” <i>Journal
    de Theorie Des Nombres de Bordeaux</i>, vol. 37, no. 3, Université de Bordeaux,
    2025, pp. 973–88, doi:<a href="https://doi.org/10.5802/jtnb.1348">10.5802/jtnb.1348</a>.
  short: Y. Diao, Journal de Theorie Des Nombres de Bordeaux 37 (2025) 973–988.
corr_author: '1'
das_tickbox: '0'
date_created: 2025-12-21T23:01:35Z
date_published: 2025-11-27T00:00:00Z
date_updated: 2026-07-16T08:45:12Z
day: '27'
ddc:
- '510'
department:
- _id: TiBr
doi: 10.5802/jtnb.1348
external_id:
  arxiv:
  - '2408.03774'
file:
- access_level: open_access
  checksum: 67aa0afbc0b5bcbff5341f4d25e6ba20
  content_type: application/pdf
  creator: dernst
  date_created: 2025-12-29T10:05:22Z
  date_updated: 2025-12-29T10:05:22Z
  file_id: '20861'
  file_name: 2025_JTNB_Diao.pdf
  file_size: 766196
  relation: main_file
  success: 1
file_date_updated: 2025-12-29T10:05:22Z
fulldoi: https://doi.org/10.5802/jtnb.1348
has_accepted_license: '1'
intvolume: '        37'
issue: '3'
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nd/4.0/
month: '11'
oa: 1
oa_version: Published Version
page: 973-988
publication: Journal de theorie des nombres de Bordeaux
publication_identifier:
  eissn:
  - 2118-8572
  issn:
  - 1246-7405
publication_status: published
publisher: Université de Bordeaux
quality_controlled: '1'
researchdata_availability: no
scopus_import: '1'
status: public
supplementarymaterial: no
title: Class numbers and integer points on some Pellian surfaces
tmp:
  image: /image/cc_by_nd.png
  legal_code_url: https://creativecommons.org/licenses/by-nd/4.0/legalcode
  name: Creative Commons Attribution-NoDerivatives 4.0 International (CC BY-ND 4.0)
  short: CC BY-ND (4.0)
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 37
year: '2025'
...
---
_id: '12406'
abstract:
- lang: eng
  text: Let X be a sufficiently large positive integer. We prove that one may choose
    a subset S of primes with cardinality O(logX) such that a positive proportion
    of integers less than X can be represented by x2+py2 for at least one p∈S.
acknowledgement: "This article is a version the author’s master thesis at the University
  of Bonn. The author would like to thank his advisor Valentin Blomer for introducing
  the problem, and giving generous feedback and encouragement along the way, especially
  during the global pandemic.\r\nThe author thanks Edgar Assing for his lectures on
  analytic number theory. Finally, the author is grateful to the anonymous referees
  for their valuable time and comments.\r\n"
article_processing_charge: No
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. Density of the union of positive diagonal binary quadratic forms. <i>Acta
    Arithmetica</i>. 2023;207:1-17. doi:<a href="https://doi.org/10.4064/aa210830-24-11">10.4064/aa210830-24-11</a>
  apa: Diao, Y. (2023). Density of the union of positive diagonal binary quadratic
    forms. <i>Acta Arithmetica</i>. Instytut Matematyczny. <a href="https://doi.org/10.4064/aa210830-24-11">https://doi.org/10.4064/aa210830-24-11</a>
  chicago: Diao, Yijie. “Density of the Union of Positive Diagonal Binary Quadratic
    Forms.” <i>Acta Arithmetica</i>. Instytut Matematyczny, 2023. <a href="https://doi.org/10.4064/aa210830-24-11">https://doi.org/10.4064/aa210830-24-11</a>.
  ieee: Y. Diao, “Density of the union of positive diagonal binary quadratic forms,”
    <i>Acta Arithmetica</i>, vol. 207. Instytut Matematyczny, pp. 1–17, 2023.
  ista: Diao Y. 2023. Density of the union of positive diagonal binary quadratic forms.
    Acta Arithmetica. 207, 1–17.
  mla: Diao, Yijie. “Density of the Union of Positive Diagonal Binary Quadratic Forms.”
    <i>Acta Arithmetica</i>, vol. 207, Instytut Matematyczny, 2023, pp. 1–17, doi:<a
    href="https://doi.org/10.4064/aa210830-24-11">10.4064/aa210830-24-11</a>.
  short: Y. Diao, Acta Arithmetica 207 (2023) 1–17.
corr_author: '1'
date_created: 2023-01-26T21:17:04Z
date_published: 2023-01-09T00:00:00Z
date_updated: 2024-10-21T06:01:30Z
day: '09'
department:
- _id: GradSch
doi: 10.4064/aa210830-24-11
external_id:
  arxiv:
  - '2103.08268'
  isi:
  - '000912903000001'
fulldoi: https://doi.org/10.4064/aa210830-24-11
intvolume: '       207'
isi: 1
keyword:
- Algebra
- Number Theory
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.48550/arXiv.2103.08268
month: '01'
oa: 1
oa_version: Preprint
page: 1-17
publication: Acta Arithmetica
publication_identifier:
  eissn:
  - 1730-6264
  issn:
  - 0065-1036
publication_status: published
publisher: Instytut Matematyczny
quality_controlled: '1'
scopus_import: '1'
status: public
title: Density of the union of positive diagonal binary quadratic forms
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 207
year: '2023'
...