On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms

Lü, Guangshi

Canadian journal of mathematics, Tome 63 (2011) no. 3, pp. 634-647

Résumé

Let ${{S}_{k}}(\Gamma )$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group. Let ${{\lambda }_{f}}(n)$ and ${{\lambda }_{g}}(n)$ be the $n$ -th normalized Fourier coefficients of two holomorphic Hecke eigencuspforms $f(z),\,g(z)\,\in \,{{S}_{k}}(\Gamma )$ , respectively. In this paper we are able to show the following results about higher moments of Fourier coefficients of holomorphic cusp forms.(i)For any $\varepsilon \,>\,0$ , we have $$\sum\limits_{n\le x}{\lambda _{f}^{5}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{15}{16}+\varepsilon }}\text{and}\sum\limits_{n\le x}{\lambda _{f}^{7}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{63}{64}+\varepsilon }}.$$ (ii)If $\text{sy}{{\text{m}}^{3\,}}{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{3\,}}{{\pi }_{g}}\,$ , then for any $\varepsilon \,>\,0$ , we have $$\sum\limits_{n\le x}{\lambda _{f}^{3}(n)\lambda _{g}^{3}(n){{\ll }_{f,\varepsilon }}}{{x}^{\frac{31}{32}+\varepsilon }};$$ If $\text{sy}{{\text{m}}^{2}}\,{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{2}}\,{{\pi }_{g}}$ , then for any $\varepsilon \,>\,0$ , we have $$\sum\limits_{n\le x}{\lambda _{f}^{4}(n)\lambda _{g}^{2}(n)}=cx\log x+{c}'x+{{O}_{f,\varepsilon }}({{x}^{\frac{31}{32}+\varepsilon }});$$ If $\text{sy}{{\text{m}}^{2}}\,{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{2}}\,{{\pi }_{g}}$ and $\text{sy}{{\text{m}}^{4}}{{\pi }_{f}}\,\ncong \,\text{sy}{{\text{m}}^{4}}{{\pi }_{g}}$ , then for any $\varepsilon \,>\,0$ , we have $$\sum\limits_{n\le x}{\lambda _{f}^{4}(n)\lambda _{g}^{4}(n)}=xP(\log x)+{{O}_{f,\varepsilon }}({{x}^{\frac{127}{128}+\varepsilon }}),$$ where $P\left( x \right)$ is a polynomial of degree 3.

Détail

DOI : 10.4153/CJM-2011-010-5

Mots-clés : 11F30, 11F11, 11F66, Fourier coefficients of cusp forms, symmetric power L-function