Pair Correlation of Zeros of Dirichlet L-Functions.

Abstract

We investigate the pair correlation function of the zeros of Dirichlet L-functions, namely F((alpha)) = F((alpha),Q) (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) and L( 1/2 + i(gamma),(chi)) = L( 1/2 + i(gamma)',(chi)) = 0. Assuming the Generalized Riemann Hypothesis, we obtain the following results: Theorem. For real (alpha), F((alpha)) is real, even and non-negative. We have (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) uniformly as Q (--->) (INFIN). Here (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) Corollary 1. If 1 (LESSTHEQ) (alpha) < 2 is fixed, then (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) Corollary 2. We have, as Q tends to infinity, (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) Corollary 2 is equivalent to the assertion that at least 11/12 of all the zeros of all Dirichlet L-functions are simple. We employ the Hardy-Littlewood circle method and some estimates on trigonometrical sums over primes to estimate expressions of the form (DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI) where a(t) is of bounded variation with compact support and (LAMDA)(n) is the von Mangoldt function. This provides an improved version of a theorem of Hooley on the mean square error in the prime number theorem for arithmetic progressions.