Pair correlation of zeros of Dirichlet L-functions: a possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery

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 $$\chi (\text {mod}\,q)$$ χ ( mod q ) for which $$L(\frac{1}{2},\chi )=0$$ L ( 1 2 , χ ) = 0 is of order less than $$q^{1/2+\varepsilon }$$ q 1 / 2 + ε .