Estimates for coefficients of certain L-functions

Abstract

Let $$\pi _{\varphi }$$ (or $$\pi _{\psi }$$ ) be an automorphic cuspidal representation of $$\text {GL}_{2} (\mathbb {A}_{\mathbb {Q}})$$ associated to a primitive Maass cusp form $$\varphi $$ (or $$\psi $$ ), and $$\mathrm{sym}^j \pi _{\varphi }$$ be the jth symmetric power lift of $$\pi _{\varphi }$$ . Let $$a_{\mathrm{sym}^j \pi _{\varphi }}(n)$$ denote the nth Dirichlet series coefficient of the principal L-function associated to $$\mathrm{sym}^j \pi _{\varphi }$$ . In this paper, we study first moments of Dirichlet series coefficients of automorphic representations $$\mathrm{sym}^3 \pi _{\varphi }$$ of $$\text {GL}_{4}(\mathbb {A}_{\mathbb {Q}})$$ , and $$\pi _{\psi }\otimes \mathrm{sym}^2 \pi _{\varphi }$$ of $$\text {GL}_{6}(\mathbb {A}_{\mathbb {Q}})$$ . For $$3 \le j \le 8$$ , estimates for $$|a_{\mathrm{sym}^j \pi _{\varphi }}(n)|$$ on average over a short interval have also been established.