  Assuming the Generalized Riemann Hypothesis (GRH), we show using the
asymptotic large sieve that 91% of the zeros of primitive Dirichlet
$L$-functions are simple. This improves on earlier work of \"{O}zl\"{u}k which
gives a proportion of at most 86%. We further compute an $q$-analogue of the
Pair Correlation Function $F(\alpha)$ averaged over all primitive Dirichlet
$L$-functions in the range $|\alpha| < 2$ . Previously such a result was
available only when the average included all the characters $\chi$.
