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Equivalent criterion for the grand Riemann hypothesis associated with Maass cusp forms

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

Published online by Cambridge University Press:  19 July 2023

Soumyarup Banerjee  and
Rahul Kumar
Soumyarup Banerjee
Affiliation:
Department of Mathematics and Statistics, Indian Institute of Science Education and Research Kolkata, Mohanpur 741246, West Bengal, India (soumya.tatan@gmail.com)
Rahul Kumar*
Affiliation:
Center for Geometry and Physics, Institute for Basic Science (IBS), Pohang 37673, Republic of Korea (rjk6031@psu.edu)
*
*Current address: Department of Mathematics, The Pennsylvania State University, University Park, PA 16802, USA.
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Abstract

In this article, we obtain transformation formulas analogous to the identity of Ramanujan, Hardy and Littlewood in the setting of primitive Maass cusp form over the congruence subgroup $\Gamma _0(N)$ and also provide an equivalent criterion of the grand Riemann hypothesis for the $L$-function associated with the primitive Maass cusp form over $\Gamma _0(N)$.

Keywords

Maass cusp formgrand Riemann hypothesisL-function associated with Maass cusp formequivalent criterionBessel function

MSC classification

Primary: 11F30: Fourier coefficients of automorphic forms 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses
Secondary: 11F12: Automorphic forms, one variable 33C10: Bessel and Airy functions, cylinder functions, $_0F_1$
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 16
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Copyright
Copyright © The Author(s), 2023. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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