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

TL;DR

This paper explores the implications of the pair correlation of zeros of Dirichlet L-functions, assuming the Generalized Riemann Hypothesis, to support major conjectures in number theory like Montgomery's, Elliott-Halberstam, and Chowla.

Contribution

It demonstrates that under certain hypotheses, key conjectures about prime distribution and zeros of L-functions can be proven or supported.

Findings

Proves Montgomery's conjecture on prime number theorem error term under assumptions.

Supports the Elliott-Halberstam conjecture assuming the pair correlation conjecture.

Shows the number of zeros of Dirichlet L-functions at 1/2 is less than q^{1/2+ε}.

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