  We use the asymptotic large sieve, developed by the authors, to prove that if
the Generalized Riemann Hypothesis is true, then there exist many Dirichlet
L-functions that have a pair of consecutive zeros closer together than 0.37
times their average spacing. More generally, we investigate zero spacings
within the family of twists by Dirichlet characters of a fixed L-function and
give precise bounds for small gaps which depend only on the degree of the
L-function.
