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

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