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 $\chi \pmod{ q}$ for which $L(\frac{ 1}{ 2},\chi)=0$ is of order less than $q^{ 1/2+\epsilon}$.
Categories: math.NT