Asymptotics for sums of twisted L-functions and applications

摘要

Let n > 2, let F be a global field containing a full set of rath roots of unity, and let TT be an isobaric automorphic representation of GLT(AF) We establish asymptotic estimates for the sum of the fi-th order twisted L-functions of it, L(s,TTx)t for s such that Re(s) > max(l -1/r, 1/2) if n = 2 and Re(s) > 1 - l/(r + 1) if n > 2. As an application we establish new nonvanishing theorems for twists of given order, including a simultaneous nonvanishing result When n = 2 we use this information on asymptotics to prove that the twisted L-values at s = 1 give rise to a distribution function.