A note on simple zeros related to Dedekind zeta functions

TL;DR

This paper establishes a conditional lower bound on the count of simple zeros of Dedekind zeta functions for quadratic fields, assuming a strong Lindel"of hypothesis, advancing previous results in the field.

Contribution

It provides a new conditional lower bound on simple zeros of Dedekind zeta functions under a stronger Lindel"of hypothesis assumption, improving prior work.

Findings

Conditional lower bound on simple zeros for quadratic Dedekind zeta functions

Assumes a strong Lindel"of hypothesis for $ ext{L}^6$ averages

Improves upon previous results by Wu and Zhao

Abstract

We give a conditional lower bound on the number of non-trivial simple zeros for the Dedekind zeta function $\zeta_{K}(s)$, where $K$ is a quadratic number field. The conditional result is given by assuming a Lindel\"of on average (in the $L^{6}$ sense) for both $\zeta(s)$ and $L(s,\chi)$, which can be seen as a stronger version of Conrey-Gonek-Ghosh's \cite{c} conditional result. This improves upon the work of Wu and Zhao \cite{Zhao}, who had a similar result.