Pair Correlation of zeros of Dirichlet $L$-Functions: A possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery

Assuming the Generalized Riemann Hypothesis and a pair correlation conjecture for the zeros of Dirichlet $L$-functions, we establish the truth of a conjecture of Montgomery (in its corrected form stated by Friedlander and Granville) on the magnitude of the error term in the prime number theorem in arithmetic progressions. As a consequence, we obtain that, under the same assumptions, the Elliott-Halberstam conjecture holds true. As another consequence, under the same assumptions, we will show that the number of Dirichlet characters $\chi (\mathrm{m}\mathrm{o}\mathrm{d}q)$ for which $L(\frac{1}{2},\chi )=0$ is of order less than $q^{1\mathrm{/}2+\epsilon }$.