---
OA_place: publisher
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abstract:
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text: Let N(X) be the number of integral zeros (mathematical equation). Works of
Hooley and Heath-Brown imply (mathematical equation), if one assumes automorphy
and grand Riemann hypothesis for certain Hasse–Weil L-functions. Assuming instead
a natural large sieve inequality, we recover the same bound on N(X). This is part
of a more general statement, for diagonal cubic forms in (mathematical equation)
variables, where we allow approximations to Hasse–Weil L-functions.
acknowledgement: I thank Peter Sarnak for suggesting projects that ultimately led
to the present paper. I also thank him for many encouraging discussions, helpful
comments, and references. Thanks also to Tim Browning, Trevor Wooley, and Nina Zubrilina
for helpful comments, and to Levent Alpöge and Will Sawin for some interesting old
discussions. I thank Yang Liu, Evan O'Dorney, Ashwin Sah, and Mark Sellke for conversations
illuminating the combinatorics of an older, counting version of the present Lemma
4.9. Finally, special thanks are due to the editors and referees for their patience
and help with the exposition. This work was partially supported by NSF Grant DMS-1802211,
and the European Union's Horizon 2020 research and innovation program under the
Marie Skłodowska-Curie Grant Agreement No. 101034413.
article_number: e70008
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author:
- first_name: Victor
full_name: Wang, Victor
id: 76096395-aea4-11ed-a680-ab8ebbd3f1b9
last_name: Wang
orcid: 0000-0002-0704-7026
citation:
ama: Wang V. Diagonal cubic forms and the large sieve. Mathematika. 2025;71(1).
doi:10.1112/mtk.70008
apa: Wang, V. (2025). Diagonal cubic forms and the large sieve. Mathematika.
London Mathematical Society. https://doi.org/10.1112/mtk.70008
chicago: Wang, Victor. “Diagonal Cubic Forms and the Large Sieve.” Mathematika.
London Mathematical Society, 2025. https://doi.org/10.1112/mtk.70008.
ieee: V. Wang, “Diagonal cubic forms and the large sieve,” Mathematika, vol.
71, no. 1. London Mathematical Society, 2025.
ista: Wang V. 2025. Diagonal cubic forms and the large sieve. Mathematika. 71(1),
e70008.
mla: Wang, Victor. “Diagonal Cubic Forms and the Large Sieve.” Mathematika,
vol. 71, no. 1, e70008, London Mathematical Society, 2025, doi:10.1112/mtk.70008.
short: V. Wang, Mathematika 71 (2025).
corr_author: '1'
date_created: 2025-01-12T23:04:01Z
date_published: 2025-01-02T00:00:00Z
date_updated: 2025-04-14T07:54:56Z
day: '02'
ddc:
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doi: 10.1112/mtk.70008
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grant_number: '101034413'
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publication: Mathematika
publication_identifier:
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issn:
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publisher: London Mathematical Society
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title: Diagonal cubic forms and the large sieve
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---
OA_place: publisher
OA_type: hybrid
_id: '17323'
abstract:
- lang: eng
text: We investigate strong divisibility sequences and produce lower and upper bounds
for the density of integers in the sequence that only have (somewhat) large prime
factors. We focus on the special cases of Fibonacci numbers and elliptic divisibility
sequences, discussing the limitations of our methods. At the end of the paper,
there is an appendix by Sandro Bettin on divisor closed sets that we use to study
the density of prime terms that appear in strong divisibility sequences.
acknowledgement: The authors are very grateful to Andrew Granville, Dimitris Koukoulopoulos,
Davide Lombardo,Florian Luca, Igor Shparlinski and Joni Teräväinen for useful comments.
While working on thispaper, the first author was supported by a FWF Grant (DOI 10.55776/P36278)
and the secondauthor was supported by the European Union’s Horizon 2020 research
and innovation programunder the Marie Skłodowska-Curie Grant Agreement Number 101034413.
article_number: e12269
article_processing_charge: Yes (via OA deal)
article_type: original
author:
- first_name: Timothy D
full_name: Browning, Timothy D
id: 35827D50-F248-11E8-B48F-1D18A9856A87
last_name: Browning
orcid: 0000-0002-8314-0177
- first_name: Matteo
full_name: Verzobio, Matteo
id: 7aa8f170-131e-11ed-88e1-a9efd01027cb
last_name: Verzobio
orcid: 0000-0002-0854-0306
citation:
ama: Browning TD, Verzobio M. Strong divisibility sequences and sieve methods. Mathematika.
2024;70(4). doi:10.1112/mtk.12269
apa: Browning, T. D., & Verzobio, M. (2024). Strong divisibility sequences and
sieve methods. Mathematika. London Mathematical Society. https://doi.org/10.1112/mtk.12269
chicago: Browning, Timothy D, and Matteo Verzobio. “Strong Divisibility Sequences
and Sieve Methods.” Mathematika. London Mathematical Society, 2024. https://doi.org/10.1112/mtk.12269.
ieee: T. D. Browning and M. Verzobio, “Strong divisibility sequences and sieve methods,”
Mathematika, vol. 70, no. 4. London Mathematical Society, 2024.
ista: Browning TD, Verzobio M. 2024. Strong divisibility sequences and sieve methods.
Mathematika. 70(4), e12269.
mla: Browning, Timothy D., and Matteo Verzobio. “Strong Divisibility Sequences and
Sieve Methods.” Mathematika, vol. 70, no. 4, e12269, London Mathematical
Society, 2024, doi:10.1112/mtk.12269.
short: T.D. Browning, M. Verzobio, Mathematika 70 (2024).
corr_author: '1'
date_created: 2024-07-28T22:01:08Z
date_published: 2024-10-01T00:00:00Z
date_updated: 2025-09-08T08:44:11Z
day: '01'
ddc:
- '510'
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doi: 10.1112/mtk.12269
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month: '10'
oa: 1
oa_version: Published Version
project:
- _id: bd8a4fdc-d553-11ed-ba76-80a0167441a3
grant_number: P36278
name: Rational curves via function field analytic number theory
- _id: fc2ed2f7-9c52-11eb-aca3-c01059dda49c
call_identifier: H2020
grant_number: '101034413'
name: 'IST-BRIDGE: International postdoctoral program'
publication: Mathematika
publication_identifier:
eissn:
- 2041-7942
issn:
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publisher: London Mathematical Society
quality_controlled: '1'
scopus_import: '1'
status: public
title: Strong divisibility sequences and sieve methods
tmp:
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legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
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type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
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...