A note on Dedekind zeta values at 1/2

TL;DR

This paper investigates the non-vanishing and transcendence properties of Dedekind zeta functions and their derivatives at s=1/2, providing new insights into their behavior and extending previous results in number theory.

Contribution

It advances understanding of Dedekind zeta values at 1/2 by establishing non-vanishing and transcendence results, strengthening prior findings by Ram Murty and Tanabe.

Findings

Proves non-vanishing of $zeta_K$ at s=1/2 for certain fields

Establishes transcendence of $zeta_K$ and its derivative at s=1/2

Strengthens previous results on the nature of $e^{b3}$ and non-vanishing of L-series

Abstract

For a number field $K$, let $\zeta_{K}(s)$ be the Dedekind zeta function associated to $K$. In this note, we study non-vanishing and transcendence of $\zeta_{K}$ as well as its derivative $\zeta_{K}'$ at $s= 1/2$. En route, we strengthen a result proved by Ram Murty and Tanabe [On the nature of $e^{\gamma}$ and non-vanishing of $L$-series at $s= 1/2$, J. Number Theory 161 (2016) 444-456].