Conditional estimates on the argument of Dirichlet L-functions with applications to low-lying zeros

Under the generalized Riemann hypothesis, we use Beurling-Selberg extremal functions to bound the mean and mean square of the argument of Dirichlet L-functions to a large prime modulus q. As applications, we give alternative proofs of several results on low-lying zeros of L (s, χ) and obtain a new lower bound on the proportion of L (s, χ) modulo q with zeros close to the central point. In particular, we show conditionally that for any β>1/4, there exist a positive proportion of Dirichlet L-functions whose first zero has height less than β times the average spacing between consecutive zeros.