On the fourth moment of coefficients of symmetric square L-function

Abstract

Let f(z) be a holomorphic Hecke eigencuspform of weight k for the full modular group. Let λ _f(n) be the nth normalized Fourier coefficient of f(z). Suppose that L(sym² f, s) is the symmetric square L-function associated with f(z), and $\lambda _{sym^2 f} (n)$ (n) denotes the nth coefficient L(sym² f, s). In this paper, it is proved that $\sum\limits_{n \leqslant x} {\lambda _{sym^2 f}^4 (n)} = xP2(\log x) + O(x^{\frac{{79}}{{81}} + \varepsilon } ),$, where P ₂(t) is a polynomial in t of degree 2. Similarly, it is obtained that $\sum\limits_{n \leqslant x} {\lambda _f^4 (n^2 )} = x\tilde P2(\log x) + O(x^{\frac{{79}}{{81}} + \varepsilon } ),$, where $\tilde P_2 (t)$ is a polynomial in t of degree 2.

Keywords

Fourier coefficient of cusp form / Symmetric power L-function / Rankin-Selberg L-function

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