Lower Bound For The Higher Moment Of Twisted L-functions

Moments of L-functions is the important research subject of analytic number theory.According to the work of Keeting et al.,asymptotic formulas for higher moments of some L-functions have been established.However,limited by techniques,the asymptotic of correct order of magnitude is only established for the lower moment,and even the lower bound of the correct order of magnitude is difficult to be obtained for the higher moment.Twisted L-functions is an important kind of automorphic L-functions.In this paper,using Rudnick and Soundararajan's methods we study the lower bound problem of the higher moment at the center value of the twisted L-functions,and obtain the lower bound results for the higher moment.We have the following theorem:Theorem 1 Let f be a holomorphic cuspidal Hecke eigenform or a cuspidal Maass Hecke eigenform that satisfies the Ramanujan-Petersson conjecture for the modular group SL₂(Z).Then for any large admissible moduloqand any natural number k where ?(q) denotes the number of primitive Dirichlet characters modulo q and the asterisk restricts the summation to all primitive characters moduloq.