Uniform estimates for sums of Fourier coefficients of cusp forms

Abstract

Let k be an even positive integer and f a holomorphic Hecke eigenform of weight k with respect to the full modular group SL(2, ℕ). Let cn be the nth coefficient of the symmetric square L-function associated to f. We study the uniform bound for the sum C(x) = Σn≦xcn with respect to the weight k and establish that $$ C(x) = \sum\limits_{n \leqq x} {c_n } \ll x^{\tfrac{3} {5}} (\log x)^{\tfrac{{22}} {5}} + k^{\tfrac{3} {2}} (\log x)^5 $$ . Other similar results are also established.