Abstract 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 (mod\, q) χ (mod q) for which L (12, ) =0 L (1 2, χ) = 0 is of order less than q^1/2+ q 1 / 2 + ε.
kandhil et al. (Sun,) studied this question.