Average Behavior Of Fourier Coefficients Of Symmetric Square L-functions Over Sum Of Two Squares

Let f(z)be a holomorphic Hecke eigenform of even weight k for the full mod-ular group SL(2,Z)and denote L(s,sym2f)be the corresponding symmetric square L-function associated to f.Suppose that ?f(n)is the n th normalized Fourier coefficient of L(s,f).The summations ?n?x?f(nj)and ?n?x ?f2(nj)were studied by many authors.Futhermore,Zhai[20]studied the average behav-ior of the power sum#12 for x?1,3 ?l?8 and a,b,l?Z,but similar results were rarely given for symmetric power L-functions.Suppose that ?sym2f(n)is the n th normalized Fourier coefficient of L(s,sym2f).In this paper,we investigate the sum#12 for x?1,2 ?l?4 and a,b,l?Z,and get the following result.#12 where P2,P3,P4 are polynomials of degree 0,0,2 respectively,and(?)#12This paper includes three chapters.In Chapter 1,we introduce the background of the subjeet and the main result.In Chapter 2,we introduce some necessary lemmas.In Chapter 3,we prove the results using the lemmas given in Chapter 2.