J ul 2 00 4 Moments of L - functions , periods of cusp forms , and cancellation in additively twisted sums on GL ( n )

In a previous paper with Schmid [29] we considered the regularity of automorphic distributions for GL(2,R), and its connections to other topics in number theory and analysis. In this paper we turn to the higher rank setting, establishing the nontrivial bound ∑ n≤T an e 2π i nα = Oε(T 3/4+ε), uniformly in α ∈ R, for an the coefficients of the L-function of a cusp form on GL(3,Z)\GL(3,R). We also derive an equivalence (Theorem 7.1) between analogous cancellation statements for cusp forms on GL(n,R), and the sizes of certain period integrals. These in turn imply estimates for the second moment of cusp form L-functions.