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An approximation formula for the shifted cubic moment of automorphic L-functions in the weight aspect

Part of: Zeta and $L$-functions: analytic theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  06 September 2023

Olga Balkanova Open the ORCID record for Olga Balkanova [Opens in a new window] ,
John Brian Conrey Open the ORCID record for John Brian Conrey [Opens in a new window]  and
Dmitry Frolenkov Open the ORCID record for Dmitry Frolenkov [Opens in a new window]
Olga Balkanova*
Affiliation:
Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Street, Moscow 119991, Russia
John Brian Conrey
Affiliation:
American Institute of Mathematics, Caltech 8-32, 1200 E California Boulevard, Pasadena, CA 91125, USA e-mail: conrey@aimath.org
Dmitry Frolenkov
Affiliation:
HSE University and Steklov Mathematical Institute of Russian Academy of Sciences, 8 Gubkina Street, Moscow 119991, Russia e-mail: frolenkov@mi-ras.ru
*
e-mail: balkanova@mi-ras.ru
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Abstract

Consider the family of automorphic L-functions associated with primitive cusp forms of level one, ordered by weight k. Assuming that k tends to infinity, we prove a new approximation formula for the cubic moment of shifted L-values over this family which relates it to the fourth moment of the Riemann zeta function. More precisely, the formula includes a conjectural main term, the fourth moment of the Riemann zeta function and error terms of size smaller than that predicted by the recipe conjectures.

Keywords

L-functionscubic momentweight aspect

MSC classification

Primary: 11F11: Holomorphic modular forms of integral weight 11M99: None of the above, but in this section 33C20: Generalized hypergeometric series, $_pF_q$
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 24
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

Research of Olga Balkanova and Dmitry Frolenkov was supported by the Theoretical Physics and Mathematics Advancement Foundation “BASIS.” Research of Brian Conrey was supported in part by an FRG grant from the NSF.

References

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