Diagonal cubic forms and the large sieve
Wang V. 2025. Diagonal cubic forms and the large sieve. Mathematika. 71(1), e70008.
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Abstract
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.
Publishing Year
Date Published
2025-01-02
Journal Title
Mathematika
Publisher
London Mathematical Society
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.
Volume
71
Issue
1
Article Number
e70008
ISSN
eISSN
IST-REx-ID
Cite this
Wang V. Diagonal cubic forms and the large sieve. Mathematika. 2025;71(1). doi:10.1112/mtk.70008
Wang, V. (2025). Diagonal cubic forms and the large sieve. Mathematika. London Mathematical Society. https://doi.org/10.1112/mtk.70008
Wang, Victor. “Diagonal Cubic Forms and the Large Sieve.” Mathematika. London Mathematical Society, 2025. https://doi.org/10.1112/mtk.70008.
V. Wang, “Diagonal cubic forms and the large sieve,” Mathematika, vol. 71, no. 1. London Mathematical Society, 2025.
Wang V. 2025. Diagonal cubic forms and the large sieve. Mathematika. 71(1), e70008.
Wang, Victor. “Diagonal Cubic Forms and the Large Sieve.” Mathematika, vol. 71, no. 1, e70008, London Mathematical Society, 2025, doi:10.1112/mtk.70008.
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