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Higher Moments of Fourier Coefficients of Cusp Forms

Published online by Cambridge University Press:  20 November 2018

Guangshi Lü  and
Ayyadurai Sankaranarayanan
Guangshi Lü
Affiliation:
School of Mathematics, Shandong University, Jinan, Shandong 250100, China e-mail: gslv@sdu.edu.cn
Ayyadurai Sankaranarayanan
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Mumbai-400005, India e-mail: sank@math.tifr.res.in
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Abstract

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Let ${{S}_{k}}\left( \Gamma \right)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group $SL\left( 2,\mathbb{Z} \right)$. Let ${{\text{ }\!\!\lambda\!\!\text{ }}_{f}}\left( n \right),{{\text{ }\!\!\lambda\!\!\text{ }}_{g}}\left( n \right),{{\text{ }\!\!\lambda\!\!\text{ }}_{h}}\left( n \right)$ be the $n$-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms $f\left( z \right)\in {{S}_{{{k}_{1}}}}\left( \Gamma \right),g\left( z \right)\in {{S}_{{{k}_{2}}}}\left( \Gamma \right)$, and $h\left( z \right)\in {{S}_{{{k}_{3}}}}\left( \Gamma \right)$, respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as $\text{ }{{\lambda }_{f}}{{\left( n \right)}^{4}}{{\lambda }_{g}}{{\left( n \right)}^{2}},\text{ }{{\lambda }_{g}}{{\left( n \right)}^{6}},\text{ }{{\lambda }_{g}}{{\left( n \right)}^{2}}{{\lambda }_{h}}{{\left( n \right)}^{4}}$, and ${{\text{ }\!\!\lambda\!\!\text{ }}_{g}}{{\left( {{n}^{3}} \right)}^{2}}$ twisted by the arithmetic function ${{\text{ }\!\!\lambda\!\!\text{ }}_{f}}\left( n \right)$.

Keywords

11F3011F66Fourier coefficients of automorphic formsDirichlet seriestriple product L-functionPerron’s formula
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 58 , Issue 3 , 01 September 2015 , pp. 548 - 560
Copyright
Copyright © Canadian Mathematical Society 2015

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