Simple Zeros of the Zeta-Function of a Quadratic Number Field, II

Abstract

Let K be a fixed quadratic extension of Q and write ζ_K(s) for the Dedekind zeta-function of K, where s = σ + it. It is wellknown, and easy to prove, that the number N_K(T) of zeros of ζ_K(s) in the region 0 < σ < 1, 0 < t ≤ T satisfies

$${{\text{N}}_{\text{K}}}\left( {\text{T}} \right)\frac{{\text{T}}}{\pi } \log T$$

as T → ∞. On the other hand, not much is known about the number of $\text{N}_\text{K}^\text{*} \left( \text{T} \right)$ that are simple.