Two-point correlation function for Dirichlet L-functions

E Bogomolny 1,2 and J P Keating 3

J. Phys. A: Math. Theor. 46 (2013) 095202 (10pp)

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 → ∞ and, importantly, includes finite-E corrections. These finite-E corrections differ from those in the case of the Riemann zeta-function, obtained in Bogomolny and Keating (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.
PACS numbers: 02.10.De, 03.65.Sq, 02.10.Yn

  • 1. Laboratoire de Physique Th ́eorique et Mod`eles Statistiques, Universite Paris-Sud, Orsay, F-91405, France
  • 2 CNRS, UMR8626, Orsay, F-91405, France
  • 3 School of Mathematics, University of Bristol, Bristol BS8 1TW, UK