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| On the Fourth Moment of Coefficients of Symmetric Square L-Function |
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Citation： |
Huixue LAO.On the Fourth Moment of Coefficients of Symmetric Square L-Function[J].Chinese Annals of Mathematics B,2012,33(6):877~888 |
| Page view： 1743
Net amount： 1392 |
Authors： |
Huixue LAO; |
Foundation： |
the National Natural Science Foundation of China (Nos. 10971119, 11101249) and the Shandong Provincial Natural Science Foundation of China (No. ZR2009AQ007). |
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| Abstract： |
Let $f(z)$ be a holomorphic Hecke eigencuspform of weight $k$ for the full modular group. Let $\lambda_f(n)$ be the $n$th normalized Fourier coefficient of $f(z)$. Suppose that $L({\rm sym}^2f,s)$ is the symmetric square $L$-function associated with $f(z)$, and $\lambda_{{\rm sym}^2f}(n)$ denotes the $n$th coefficient $L({\rm sym}^2f,s)$. In this paper, it is proved that
\sum_{n \leq x}\lambda_{{\rm sym}^2f}^4(n)=xP_2(\log x)+O(x^{\frac{79}{81}+\varepsilon}),
where $P_2(t)$ is a polynomial in $t$ of degree $2$. Similarly, it is obtained that
\sum_{n \leq x}\lambda_f^4(n^2)=x \wt{P}_2(\log x)+O(x^{\frac{79}{81}+\varepsilon}),
where $\wt{P}_2(t)$ is a polynomial in $t$ of degree $2$. |
Keywords： |
Fourier coefficient of cusp form, Symmetric power L-function, Rankin-Selberg L-function |
Classification： |
11F30, 11F11, 11F66 |
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