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Oscillations of coefficients of Dirichlet series attached to automorphic forms


Authors: Jaban Meher and M. Ram Murty
Journal: Proc. Amer. Math. Soc. 145 (2017), 563-575
MSC (2010): Primary 11M41, 11M45, 11F46, 11F66
DOI: https://doi.org/10.1090/proc/13264
Published electronically: September 15, 2016
MathSciNet review: 3577861
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Abstract: For $ m\ge 2$, let $ \pi $ be an irreducible cuspidal automorphic representation of $ GL_m(\mathbb{A}_{\mathbb{Q}})$ with unitary central character. Let $ a_\pi (n)$ be the $ n^{th}$ coefficient of the $ L$-function attached to $ \pi $. Goldfeld and Sengupta have recently obtained a bound for $ \sum _{n\le x} a_\pi (n)$ as $ x \rightarrow \infty $. For $ m\ge 3$ and $ \pi $ not a symmetric power of a $ GL_2(\mathbb{A}_{\mathbb{Q}})$-cuspidal automorphic representation with not all finite primes unramified for $ \pi $, their bound is better than all previous bounds. In this paper, we further improve the bound of Goldfeld and Sengupta. We also prove a quantitative result for the number of sign changes of the coefficients of certain automorphic $ L$-functions, provided the coefficients are real numbers.


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Additional Information

Jaban Meher
Affiliation: School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar, Via-Jatni, Khurda 752050, Odisha, India
Email: jaban@niser.ac.in

M. Ram Murty
Affiliation: Department of Mathematics, Queen’s University, Kingston, Ontario, Canada K7L 3N6
Email: murty@mast.queensu.ca

DOI: https://doi.org/10.1090/proc/13264
Keywords: Automorphic $L$-functions, Siegel modular forms
Received by editor(s): October 5, 2015
Received by editor(s) in revised form: April 27, 2016
Published electronically: September 15, 2016
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2016 American Mathematical Society