Non-vanishing of high derivatives of Dirichlet L-functions at the central point

Using the mollification method. we show that, for large q, at least 2/3 - O(k(-2)) of the set {Lambda((k))(chi, 1/2)} is non-zero, where Lambda(chi, s) is the complete Dirichlet L-function and chi runs over all even primitive characters modulo q. (C) 2000 Academic Press.