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Strong orthogonality between the Möbius function, additive characters and Fourier coefficients of cusp forms

Part of: Multiplicative number theory Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  24 April 2014

Étienne Fouvry  and
Satadal Ganguly
Étienne Fouvry
Affiliation:
Laboratoire de Mathématique, Université Paris Sud, UMR 8628, CNRS, Orsay, F–91405, France email Etienne.Fouvry@math.u-psud.fr
Satadal Ganguly
Affiliation:
Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, 203 Barrackpore Trunk Road, Kolkata 700108, India email sgisical@gmail.com
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Abstract

Let $\nu _{f}(n)$ be the $n\mathrm{th}$ normalized Fourier coefficient of a Hecke–Maass cusp form $f$ for ${\rm SL }(2,\mathbb{Z})$ and let $\alpha $ be a real number. We prove strong oscillations of the argument of $\nu _{f}(n)\mu (n) \exp (2\pi i n \alpha )$ as $n$ takes consecutive integral values.

Keywords

cusp formsMöbius functionadditive characters

MSC classification

Secondary: 11F30: Fourier coefficients of automorphic forms 11N75: Applications of automorphic functions and forms to multiplicative problems
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 5 , May 2014 , pp. 763 - 797
Copyright
© The Author(s) 2014 

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