Lower bounds for small fractional moments of Dirichlet $L$-functions
  We prove a lower bound of the correct order of magnitude in the conductor aspect for small rational moments of Drichlet $L$-functions. Such bounds require new techniques, which is visible from the relationship to non-vanishing results for $L(1/2, \chi)$. 
Vorrapan Chandee, Xiannan Li
math.NT