Transcendency of the determinant of the Riemann operator: on higher $K$-groups

TL;DR

This paper explores the transcendental nature of the determinant of the Riemann operator associated with higher $K$-groups of algebraic number fields, revealing its transcendence depends on the type of the field and specific rational points.

Contribution

It establishes the transcendental properties of the determinant of the Riemann operator at certain rational points, linking algebraic number field types to transcendence results.

Findings

$G_{K}(1/3)$ is transcendental if $K$ is totally imaginary.

$G_{K}(1/2)$ is transcendental for other types of $K$.

The study connects the determinant's transcendence to properties of algebraic number fields.

Abstract

In previous papers we investigated basic properties of the determinant $G_{K}(s)$ of the Riemann operator: ${\mathcal R}$ acting on $\bigoplus_{n>1} K_{n}(A)_{\mathbb{C}}$, where $A$ is the integer ring of an algebraic number field $K$. The function $G_{K}(s)$ is defined as the regularized determinant \[ G_{K}(s) = {\rm det} ((s I-\mathcal{R}) | \bigoplus_{n>1} K_{n}(A)_{\mathbb{C}} ) \] with $\mathcal{R} | K_{n}(A)_{\mathbb{C}} = \frac{1-n}{2}$. We showed that $G_{K}(s)^{-1}$ is essentially the so called gamma factors of Dedekind zeta function of $K$. In this paper we study the transcendency of $G_{K}(s)$ for some rational numbers $s$. The result depends on types of $K$. For example, we show that $G_{K}(\frac{1}{3})$ is a transcendental number if $K$ is a totally imaginary and $G_{K}(\frac{1}{2})$ is a transcendental number otherwise.