On Higher Moments of Fourier Coefficients of Holomorphic Cusp Forms

Let S(k)(Gamma) be the space of holomorphic cusp forms of even integral weight k for the full modular group. Let lambda(f) (n) and lambda(g)(n) be the n-th normalized Fourier coefficients of two holomorphic Hecke eigencuspforms f (z), g(z) is an element of S(k)(Gamma), respectively. In this paper we are able to show the following results about higher moments of Fourier coefficients of holomorphic cusp forms. (i) For any epsilon > 0, we have Sigma n <= x lambda(5)(f)(n) << f,epsilon x(15/16+epsilon) and Sigma n <= x lambda(7)(f)(n) << f,epsilon x(63/64+epsilon) (ii) If sym(3) pi(f) not congruent to sym(3) pi(g), then for any epsilon > 0, we have Sigma n <= x lambda(3)(f)(n)lambda(3)(g)(n) << f,epsilon x(31/32+epsilon) ; In sym(2) pi(f) not congruent to sym(2) pi(g), then for any epsilon > 0, we have Sigma n <= x lambda(4)(f)(n) lambda(2)(g) (n) = cx log x + c'x + O(f,epsilon) x(31/32+)epsilon ; In sym(2) pi(f) not congruent to sym(2) pi(g) and sym(4) pi(f) not congruent to sym(4) pi(g) then for any epsilon > 0, we have Sigma n <= x lambda(4)(f)(n) lambda(4)(g)(n) = zP(log x) + P(f,epsilon) x(127/128+epsilon) , where P(x) is a polynomial of degree 3.