论文标题
Riemann运营商决定因素的超越：在高级$ k $ -groups上
Transcendency of the determinant of the Riemann operator: on higher $K$-groups
论文作者
论文摘要
在先前的论文中，我们调查了Riemann操作员的行列式$ g_ {k}（k}（s）$：$ {\ Mathcal r} $作用于$ \ bigoplus_ {n> 1} k_ {n>} k_ {n}（n}（n} n}（a）_ {\ mathbb {c} c} $ kn $ kn $ kn $ kn $ kn $函数$ g_ {k}（s）$定义为正规化的确定性\ [g_ {k {k}（s）= {\ rm det}（（（s i- \ mathcal {r}）| \ bigoplus_ {n> 1} k_ {n} k_ {n}（a） $ \ mathcal {r} | k_ {n}（a）_ {\ mathbb {c}} = \ frac {1-n} {2} $。我们表明$ g_ {k}（s）^{ - 1} $本质上是$ k $的Dedekind Zeta函数的所谓伽马因子。在本文中，我们研究了某些合理数字$ s $的$ g_ {k}（s）$的超越性。结果取决于$ k $的类型。例如，我们表明$ g_ {k}（\ frac {1} {3}）$是一个先验数，如果$ k $是一个完全虚构的，而$ g_ {k {k {k {\ frac {1} {2} {2} {2}）$是一个超定号的数字。
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.
