let $L(s,\chi)$ be its associated Dirichlet L-function. Under GRH, all non-trivial zeros of $L(s,\chi)$ lie on the critical line ${{\rm Re}\,}s = 1/2$. Let $$N_{\Phi}(Q):=\displaystyle\sum_{Q\leq q\leq 2Q}\dfrac{W(q/Q)}{\varphi(q)}\displaystyle\sum_{\substack{\chi  \ (\mathrm{mod} \ q) \\ \text{primitive}}} \displaystyle\;\sum_{\gamma_\chi}\big|\mathcal{M}\Phi(i\gamma_\chi)\big|^2,$$ where $W$ is a non-negative smooth function supported in $(1,2)$, and where the last sum is over all non-trivial zeros ${\tfrac12}+i\gamma_\chi$ of the Dirichlet $L$-function $L(s,\chi)$. In [@CLLR Lemma 2.1] it is shown that $$N_{\Phi}(Q)\sim\dfrac{A}{2\pi}Q\log Q \int_{-\infty}^{\infty}\big|\mathcal{M}\Phi(ix)\big|^2{\,\text{\rm d}}x,$$ where $$A={\mathcal{M}}W(1)\displaystyle\prod_{p\ \text{prime}}\bigg(1-\dfrac{1}{p^2}-\dfrac{1}{p^3}\bigg).$$ Let $$N_{\Phi,s}(Q)=\displaystyle\sum_{Q\leq q\leq 2Q}\dfrac{W(q/Q)}{\varphi(q)}\displaystyle\sum_{\substack{\chi  \ (\mathrm{mod} \ q) \\ \text{primitive}}} \displaystyle\;\sum_{\substack{\gamma_\chi \\ \text{simple}}}\big|\mathcal{M}\Phi(i\gamma_\chi)\big|^2.$$ The quantity $$\liminf_{Q\to\infty}\frac{N_{\Phi,s}(Q)}{N_{\Phi}(Q)}$$ then measures (in average) the proportion of simple zeros among all primitive Dirichlet L-functions.

In addition, for the following theorem, we require that $\Phi(x)$ and ${\mathcal{M}}\Phi(ix)$ are non-negative functions. We note that we can also further relax the conditions on $\Phi$ so to include the function given by ${\mathcal{M}}\Phi(ix) = (\sin x/x)^2$, as was established in [@CLLR] and [@S].

\[thmsimplezerosofdir\] Assuming GRH we have $$N_{\Phi,s}(Q)\geq (0.9350+o(1))N_{\Phi}(Q).$$

Using the pair correlation approach, the best previous result known is due to Sono [@S], showing that $93.22\%$ of the zeros are simple. To the best of our knowledge, our bound is the current best.

Results for zeros of $\xi'(s)$
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We can 