Two-point correlation function for Dirichlet L-functions

TL;DR

This paper heuristically derives the two-point correlation function for zeros of Dirichlet L-functions, matching random matrix theory predictions and including finite-energy corrections that depend on the modulus of the character.

Contribution

It extends the understanding of zero correlations for Dirichlet L-functions by incorporating finite-energy corrections and generalizing previous results for the Riemann zeta-function.

Findings

Correlation function matches random matrix theory predictions as E→∞

Finite-E corrections depend on primes dividing the modulus

Results differ from zeta-function case by specific prime factors

Abstract

The two-point correlation function for the zeros of Dirichlet L-functions at a height E on the critical line is calculated heuristically using a generalization of the Hardy-Littlewood conjecture for pairs of primes in arithmetic progression. The result matches the conjectured Random-Matrix form in the limit as $E\rightarrow\infty$ and, importantly, includes finite-E corrections. These finite-E corrections differ from those in the case of the Riemann zeta-function, obtained in (1996 Phys. Rev. Lett. 77 1472), by certain finite products of primes which divide the modulus of the primitive character used to construct the L-function in question.