Bulletin of the Australian Mathematical Society

Research Article

Small zeros of quadratic L-functions

Ali E. Özlüka1 and C. Snydera1

a1 Department of Mathematics, University of Maine, Neville Hall Orono ME 04469 0122, United States of America

Abstract

We study the distribution of the imaginary parts of zeros near the real axis of quadratic L-functions. More precisely, let K(s) be chosen so that |K(1/2 ± it)| is rapidly decreasing as t increases. We investigate the asymptotic behaviour of

S0004972700012545_eqnU1

as D → ∞. Here S0004972700012545_inline1 denotes the sum over the non-trivial zeros p = 1/2 + of the Dirichlet L-function L(s, χd), and χd = (S0004972700012545_inline2) is the Kronecker symbol. The outer sum S0004972700012545_inline3 is over all fundamental discriminants d that are in absolute value ≤ D. Assuming the Generalized Riemann Hypothesis, we show that for

S0004972700012545_eqnU2

(Received April 14 1992)