and Lee [@FGL] with constant $0.8051$. By a different technique, assuming RH, Bui and Heath-Brown [@BHB] improved the constant to $0.8466$. To the best of our knowledge, our new bounds are the current best.

We also obtain improved results for Montgomery’s pair correlation function.

\[thmpaircorreltation\] Assuming RH and we have $$N({0.6039},T)\gg N(T).$$ Assuming GRH and we have $$N(0.5769,T)\gg N(T).$$

Montgomery [@M] showed that $N(0.68..., T)\gg N(T)$, and in [@GGOS] it is pointed out that it is not difficult to modify Montgomery’s argument to derive the sharper constant $0.6695$. This result was improved by Goldston, Gonek, Özlük and Snyder [@GGOS] with constant $0.6072$. Later, Carneiro, Chandee, Littmann and Milinovich [@CCLM] improved the constant to $0.6068...$. Assuming GRH and , Goldston, Gonek, Özlük and Snyder showed the constant $0.5781...$. To the best of our knowledge, our new bounds are the current best.

Results for zeros of Dirichlet L-functions
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To obtain averaged bounds for the percentage of simple zeros of primitive Dirichlet L-functions we use the framework established by Chandee, Lee, Liu and Radziwiłł [@CLLR]. Let $\Phi$ be a real-valued smooth function supported in the interval $[a,b]$ with $0<a<b<\infty$. Define its Mellin transform by $$\mathcal{M}\Phi(s)=\int_{0}^{\infty}\Phi(x)x^{s-1}{\,\text{\rm d}}x.$$ For a character $\chi$ mod $q$, 