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Cancellation of two classes of dirichlet coefficients over Beatty sequences

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

Published online by Cambridge University Press:  20 April 2021

Qiang Ma
Qiang Ma*
Affiliation:
School of Mathematics, Shandong University, Jinan250100, China
*
e-mail: qma@mail.sdu.edu.cn
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Abstract

Let $\pi $ be an automorphic irreducible cuspidal representation of $\mathrm{GL}_{m}$ over $\mathbb {Q}$ . Denoted by $\lambda _{\pi }(n)$ the nth coefficient in the Dirichlet series expansion of $L(s,\pi )$ associated with $\pi $ . Let $\pi _{1}$ be an automorphic irreducible cuspidal representation of $\mathrm{SL}(2,\mathbb {Z})$ . Denoted by $\lambda _{\pi _{1}\times \pi _{1}}(n)$ the nth coefficient in the Dirichlet series expansion of $L(s,\pi _{1}\times \pi _{1})$ associated with $\pi _{1}\times \pi _{1}$ . In this paper, we study the cancellations of $\lambda _{\pi }(n)$ and $\lambda _{\pi _{1}\times \pi _{1}}(n)$ over Beatty sequences.

Keywords

Automorphic representationDirichlet coefficientsBeatty sequences

MSC classification

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11B83: Special sequences and polynomials
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022 , pp. 234 - 252
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Copyright
© Canadian Mathematical Society 2021

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