Asymptotics for sums of twisted Lfunctions and applications

This paper is dedicated to Professor Stephen J. Rallis Abstract. Let n ≥ 2, let F be a global field containing a full set of n-th roots of unity, and let pi be an isobaric automorphic representation of GLr(AF). We establish asymptotic estimates for the sum of the n-th order twisted L-functions of pi, L(s, pi ⊗ χ), for s such that Re(s)> max(1 − 1/r, 1/2) if n = 2 and Re(s)> 1 − 1/(r + 1) if n> 2. As an application we establish new non-vanishing theorems for twists of given order, including a simultaneous nonvanishing result. When n = 2 and each factor of pi is tempered we use this information on asymptotics to prove that the twisted L-values at s = 1 give rise to a distribution function. 1