Integral Power Sums Of Hecke Eigenvalues

The set of primitive holomorphic forms of even integral weight k≥2 for the full modular group SL2(Z), denoted by Hk*, consists of the common eigenfunctions f of all Hecke operators Tn, whose Fourier series expansion at the cusp oo is of form The coefficient λf(n) is the normalized Hecke eigenvalue of Tn.For f∈Hk*, we define where l∈N and x≥1.In this paper we consider the asymptotic property of Sl(f; x). Lau, Lii & Wu [28] proved where P4(t), P6(t), Ps(t) are polynomials of degree 1,4,13 respectively, Pl=0 for l=3,5,7 and The key point of their proof is the good property of L(s, symmf) in the excellent work of Kim & Shahidi [11].The author studies Rankin-Selberg method and the properties of symmetric power L-functions and makes use of expression of Ge(s) which decomposes Gl(s) into a product of L-functions, general and of low degree (see Lau, Lii,& Wu[28]). We take advantage of subconvexity bounds of the Riemann (-functions different from [28] to obtain more delicate results. For l=4,6,8, we prove the following theorem:Our paper includes four chapters. In Chapter 1 we introduce systemati-cally the background of the subject and give the main results.In Chapter 2 we give the definitions of Rankin-Selberg L-funcion, the m-th symmetric power L-function, general L-function and the related preliminary knowledge.In Chapter 3 we introduce the related lemmas. Our main idea is to de-compose Gl(s) into a product of Riemann (-function and L-functions, general and of low degree and introduce their subconvexity bounds.In Chapter 4 we take advantage of Perron’s formula and lemmas in Chap-ter 3 to prove the results.