Journal article

New large value estimates for Dirichlet polynomials

Abstract:: We prove new bounds for how often Dirichlet polynomials can take large values. This gives improved estimates for a Dirichlet polynomial of length N taking values of size close to N^3/4 , which is the critical situation for several estimates in analytic number theory connected to prime numbers and the Riemann zeta function. As a consequence, we deduce a zero density estimate N(σ, T) ≤ T^{30(1−σ)/13+o(1)} and asymptotics for primes in short intervals of length x^17/30+o(1).

Publication status:: Published

Peer review status:: Peer reviewed

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APA Style

Guth, L., & Maynard, J. (2026). New large value estimates for Dirichlet polynomials. Annals of Mathematics, 203(2), 623–675.

MLA Style

Guth, L, and J Maynard. “New Large Value Estimates for Dirichlet Polynomials.” Annals of Mathematics, vol. 203, no. 2, 2026, pp. 623–75.

Chicago Style

Guth, L, and J Maynard. 2026. “New Large Value Estimates for Dirichlet Polynomials.” Annals of Mathematics 203 (2): 623–75.
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Files:: Guth_and_Maynard_2025_New_large_value.pdf

(Preview, Accepted manuscript, pdf, 623.7KB, Terms of use)

Publisher copy:: 10.4007/annals.2026.203.2.6

Authors

+ Guth, L More by this author

Role:: Author

+ Maynard, J More by this author

Institution:: University of Oxford
Division:: MPLS
Department:: Mathematical Institute
Role:: Author

Publisher:: Princeton University
Journal:: Annals of Mathematics More from this journal
Volume:: 203
Issue:: 2
Pages:: 623-675
Publication date:: 2026-03-01
Acceptance date:: 2025-04-11
DOI:: 10.4007/annals.2026.203.2.6
EISSN:: 1939-8980
ISSN:: 0003-486X

Language:: English
Keywords:: Dirichlet polynomials

large values estimates

zero density estimates
Pubs id:: 2119163
Local pid:: pubs:2119163
Deposit date:: 2025-04-20
ARK identifier:: ark:/29072/ora_ad11b8bfad2b4ebfa627647f023c378f

Terms of use

Copyright holder:: Department of Mathematics, Princeton University
Copyright date:: 2026
Rights statement:: Copyright © 2026 Department of Mathematics, Princeton University
Notes:: The author accepted manuscript (AAM) of this paper has been made available under the University of Oxford's Open Access Publications Policy, and a CC BY public copyright licence has been applied.

Licence:: CC Attribution (CC BY)

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