The average behavior of fourier coefficients of cusp forms over sparse sequences

Abstract

Let λ(n) be the nth normalized Fourier coefficient of a holomorphic Hecke eigenform f(z) Sκ(G). In this paper we are interested in the average behavior of λ2(n) over sparse sequences. By using the properties of symmetric power L-functions and their Rankin-Selberg L-functions, we are able to establish that for any e > 0, σn≤x λ2(nj) = cj-1x + O(x 1-2(j+1)2+2 +e), where j = 2, 3, 4. © 2009 American Mathematical Society.