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arXiv · 2209.13843

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

Nobushige Kurokawa·Hidekazu Tanaka
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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.

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Nobushige Kurokawa, Hidekazu Tanaka. 2022-09-28. Determinants of Riemann operators on Quillen's higher $K$-groups: periodicity. https://arxiv.org/abs/2209.13843

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