On the coefficients of $\ell$-fold product $L$-function

Abstract

Let $f\in S_k(S{L}_2(\mathbb{Z}))$ be a normalized Hecke eigenforms of integral weight $k$ for the full modular group. In the article, we study the average behaviour of Fourier coefficients of $\ell$-fold product $L$-function. More precisely, we establish the asymptotics of power moments associated to the sequence $\{{\lambda }_{f\otimes f\otimes \cdots {\otimes }_{\ell }f}(n){\}}_{n-\mathrm{squarefree}}$ where $f\otimes f\otimes \cdots {\otimes }_{\ell }f$ denotes the $\ell$-fold product of $f$. As a consequence, we prove results concerning the behaviour of sign changes associated to these sequences for odd $\ell$-fold product $L$-function. A similar result also holds for the sequence $\{{\lambda }_{f\otimes f\otimes \cdots {\otimes }_{\ell }f}(n){\}}_{n\in \mathbb{N}}$.

13 pages, Any comments or suggestions are welcome