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

Simple zeros of primitive Dirichlet $L$-functions and the asymptotic large sieve