The number of zeros of the Dedekind zeta-function on the critical line

Erich Hecke first showed that the Dedekind zeta-function for an ideal class in an imaginary quadratic field has an infinite number of zeros on the critical line. Recently, K. Chandrasekharan and Raghavan Narasimhan proved the result for both real and imaginary quadratic fields. In this paper we give a quantitative result. Namely, let N₀(T) denote the number of zeros of the Dedekind zeta-function ζ_K( 1 2 + it T) for 0 < t < T. Then, for every ε > 0, there exists a positive constant A such that N₀(T) > AT ^{1 2-ε{lunate}}.