Higher Moments of Fourier Coefficients of Cusp Forms

Lü, Guangshi ; Sankaranarayanan, Ayyadurai

Canadian mathematical bulletin, Tome 58 (2015) no. 3, pp. 548-560

Résumé

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 ${{\text{ }\!\!\lambda\!\!\text{ }}_{f}}\left( n \right)$ .

Détail

DOI : 10.4153/CMB-2015-031-1

Mots-clés : 11F30, 11F66, Fourier coefficients of automorphic forms, Dirichlet series, triple product L-function, Perron’s formula