Pair Correlation of zeros of Dirichlet $L$-Functions: A possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery
Neelam Kandhil, Alessandro Languasco, Pieter Moree
TL;DR
This work connects the pair correlation of zeros of Dirichlet $L$-functions to major conjectures on primes by assuming GRH and a conjectural Dirichlet $L$-pair-correlation bound. The authors introduce a Dirichlet $L$-pair-correlation framework via $G_{\chi_1,\chi_2}(x,T)$ and its aggregated sums $F_q(x,T)$ and $F^+_q(x,T)$, then derive mean-square bounds and asymptotics that translate into statements about primes in arithmetic progressions. Under the conjectural bound and GRH, they prove Montgomery’s conjecture in Friedlander–Granville’s corrected form, establish Elliott–Halberstam-type bounds for the prime distribution in progressions, and show that the number of Dirichlet characters with $L(\tfrac12,\chi)=0$ is $O(q^{1/2+\varepsilon})$. The results illuminate how zeros of Dirichlet $L$-functions influence prime distribution and yield a pathway toward Chowla-type nonvanishing phenomena and Montgomery-type conjectures. The method hinges on relating zero correlations to primes via explicit formulae, controlling both high- and low-lying zeros through $F_q$-type objects, and employing a conjugation trick to handle zeros across the critical line.
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 $χ\pmod{q}$ for which $L(\frac{1}{2},χ)=0$ is of order less than $q^{1/2+ε}$.