A generalization of the Goldfeld-Sarnak estimate on Selberg's Kloosterman zeta-function

In the mid-60's Selberg [9] pointed out the importance of finding a good growth estimate on a Dirichlet series which today bears his name. Special values of this function, which involves generalized Kloosterman sums, appear in the Fourier coefficients of automorphic forms (of arbitrary real weight and multiplier system) on certain subgroups of SL(2, IR). In the early 80's Goldfeld & Sarnak [1] provided bounds which are valid to the right of the critical line sigma = 1/2. Here we extend their work by establishing estimates which hold in certain pole-free sets throughout the whole s-plane.