Determinants of Riemann operators on Quillen's higher $K$-groups: periodicity

TL;DR

This paper explores the periodicity and reflection properties of determinants of Riemann operators on Quillen's higher K-groups, linking them to classical gamma function identities and the periodicity of algebraic K-theory.

Contribution

It introduces the study of periodicity of determinants of Riemann operators on higher K-groups, connecting algebraic K-theory with classical gamma function properties.

Findings

Determinants express the inverse of gamma factors of Dedekind zeta functions.

Periodicities in K-groups mirror Euler's gamma function periodicity.

Reflection formulas analogous to Euler's reflection formula are investigated.

Abstract

In a previous paper [KT] we introduced determinant of the Riemann operator on Quillen's higher $K$-groups of the integer ring of an algebraic number field $K$. We showed that the determinant expresses essentially the inverse of the so called gamma factor of Dedekind zeta function of $K$. Here we study the periodicity of determinant. This comes from the famous "periodicity" of higher $K$ groups. This periodicity is analogous to Euler's periodicity of gamma function $\Gamma(x+1)=x\Gamma(x)$. We investigate the "reflection formula" corresponding to Euler's reflection formula $\Gamma(x)\Gamma(1-x)=\frac{\pi}{\sin(\pi x)}$ also.