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Motohashi’s formula for the fourth moment of individual Dirichlet L-functions and applications

Kaneko, Ikuya (2022) Motohashi’s formula for the fourth moment of individual Dirichlet L-functions and applications. Forum of Mathematics, Sigma, 10 . Art. No. E41. ISSN 2050-5094. doi:10.1017/fms.2022.33.

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A new reciprocity formula for Dirichlet L-functions associated to an arbitrary primitive Dirichlet character of prime modulus q is established. We find an identity relating the fourth moment of individual Dirichlet L-functions in the t-aspect to the cubic moment of central L-values of Hecke-Maass newforms of level at most q² and primitive central character ψ² averaged over all primitive nonquadratic characters ψ modulo q. Our formula can be thought of as a reverse version of recent work of Petrow-Young. Direct corollaries involve a variant of Iwaniec's short interval fourth moment bound and the twelfth moment bound for Dirichlet L-functions, which generalise work of Jutila and Heath-Brown, respectively. This work traverses an intersection of classical analytic number theory and automorphic forms.

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Kaneko, Ikuya0000-0003-4518-1805
Additional Information:© The Author(s), 2022. Published by Cambridge University Press. This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (, which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited. Received: 30 November 2021; Revised: 27 March 2022; Accepted: 24 April 2022. The author expresses his gratitude to Eren Mehmet Kıral for supervising this work. He also thanks Valentin Blomer, Jack Buttcane, Paul Nelson and Peter Sarnak for having an interest in earlier versions of this article. Appendix A is attributed to Peter Humphries, and the author acknowledges his diversified comments that improved the readability of this article. Special thanks are owed to Alexandre Perozim de Faveri, Yoichi Motohashi, Maksym Radziwiłł and Will Sawin for their assistance. Conflicts of Interest: None. The author acknowledges the support of the Masason Foundation and the Spirit of Ramanujan STEM Talent Initiative.
Funding AgencyGrant Number
Masason FoundationUNSPECIFIED
Spirit of Ramanujan STEM Talent InitiativeUNSPECIFIED
Subject Keywords:Motohashi’s formula, Dirichlet L-functions, fourth moment
Classification Code:2020 Mathematics Subject Classification: Primary – 11M06; Secondary – 11F03, 11F72
Record Number:CaltechAUTHORS:20220802-743424000
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Official Citation:Kaneko, I. (2022). Motohashi’s formula for the fourth moment of individual Dirichlet L-functions and applications. Forum of Mathematics, Sigma, 10, E41. doi:10.1017/fms.2022.33
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:116025
Deposited By: George Porter
Deposited On:03 Aug 2022 14:43
Last Modified:03 Aug 2022 14:43

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