# 데데킨트 제타함수

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## 개요

• 수체 $$K$$에 대하여, 데데킨트 제타함수는 다음과 같이 정의됨

$\zeta_{K}(s):=\sum_{I \text{:ideals}}\frac{1}{N(I)^s}$

### 기호

• $$K$$ 수체
• $$C_K$$ ideal class group

## 함수방정식

• 리만제타함수 의 함수방정식$\xi(s) : = \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$\xi(s) = \xi(1 - s)$
• 리만제타함수는 $$K=\mathbb{Q}$$ 인 경우, 즉 $$\zeta(s)=\zeta_{\mathbb{Q}}(s)$$
• 데데킨트 제타함수에 대해서 다음과 같은 함수방정식이 성립$\xi_{K}(s)=\left|d_K\right|{}^{s/2} 2^{r_2 (1-s)} \pi ^{\frac{1}{2} \left(-r_1-2 r_2\right) s}\Gamma \left(\frac{s}{2}\right)^{r_1} \Gamma (s)^{r_2}\zeta _K(s)$$\xi_{K}(s) = \xi_{K}(1 - s)$

## 디리클레 유수 공식

$\lim_{s\to 1} (s-1)\zeta_K(s)=\frac{2^{r_1}\cdot(2\pi)^{r_2}\cdot h_K\cdot R_K}{w_K \cdot \sqrt{|D_K|}}$

• $$s=0$$ 에서 order 가 $$r_1+r_2-1$$ 인 zero를 가지며 다음이 성립한다$\lim_{s\to 0}\frac{\zeta_K(s)}{s^{r_1+r_2-1}}=-\frac{h_K R_K}{w_K}$

## 부분제타함수

• 각각의 ideal class $$A\in C_K$$ 에 대하여, 부분 데데킨트 제타함수를 다음과 같이 정의$\zeta_{K}(s,A)=\sum_{\mathfrak{a} \in A }\frac{1}{N(\mathfrak{a})^s}$
• 제타함수는 부분 데데킨트 제타함수의 합으로 쓰여지게 됨$\zeta_{K}(s)=\sum_{A \in C_K}\zeta_{K}(s,A)$
• 더 일반적으로 준동형사상 $$\chi \colon C_K \to \mathbb C^{*}$$에 대하여, 일반화된 데데킨트 제타함수를 정의할 수 있음$L(\chi,s) =\sum_{\mathfrak{a} \text{:ideals}}\frac{\chi(\mathfrak{a})}{N(\mathfrak{a})^s} = \sum_{A\in C_K}{\chi(A)}\zeta_K(s,A)$

## special values

### 클링겐-지겔 (Klingen-Siegel) 정리

$\zeta_{F}(2m)=r(m)\frac{\pi^{2mn}}{\sqrt{|d_{F}|}}$

### Zagier, Bloch, Suslin

• $$[K : \mathbb{Q}] = r_1 + 2r_2$$일 때,

$\zeta_{K}(2)\sim_{\mathbb{Q^{\times}}} \frac{\pi^{2(r_1 + r_2)}}{\sqrt{|d_{K}|}}\det\{D(\sigma_i(\xi_j))\}_{1\leq i,j\leq r_2}$ 여기서 $$\xi_i,(i=1,\cdots, r_2)$$ 는 Bloch group $$B(K)\otimes \mathbb{Q}$$의 $$\mathbb{Q}$$-basis D는 블로흐-비그너 다이로그(Bloch-Wigner dilogarithm) 함수이며, $$a\sim_{\mathbb{Q^{\times}}} b$$ 는 $$a/b\in\mathbb{Q}$$ 를 의미함

## 노트

### 말뭉치

1. In particular some of these pairs have different class numbers, so the Dedekind zeta function of a number field does not determine its class number.[1]
2. For K K a number field then all special values of the Dedekind zeta function ζ K ( n ) \zeta_K(n) for integer n n happen to be periods (MO comment).[2]
3. Just like the Riemann zeta function, each Dedekind zeta function possesses a functional equation.[3]
4. The nontrivial zeros of the Dedekind zeta function of any algebraic number eld lie on the critical line: Re(s) = 1/2.[4]
5. Theorem Let X be a group of Dirichlet characters, K the associated eld, and K (s) the Dedekind zeta function of K .[4]
6. From there, we discuss algebraic number elds and introduce the tools needed to dene the Dedekind zeta function.[5]
7. 1 2 FRIMPONG A. BAIDOO necessary for providing context to the Dedekind zeta function.[5]
8. In section 9, we then dene the Dedekind zeta function, describe the ideal class group and then highlight the Dedekind zeta functions role in the class number formula.[5]
9. I was trying to learn a little about the Dedekind zeta function.[6]
10. For a cubic extension K 3 /ℚ, which is not normal, new results on the behavior of mean values of the Dedekind zeta function of the field K 3 in the critical strip are obtained.[7]
11. We study analytic aspects of the Dedekind zeta function of a Galois extension.[8]
12. In the rst part of this thesis we give a formula for the second moment of the Dedekind zeta function of a quadratic eld times an arbitrary Dirichlet polynomial of length T 1/11(cid:15).[8]
13. In the second part, we derive a hybrid Euler-Hadamard product for the Dedekind zeta function of an arbitrary number eld.[8]
14. We then conjecture that the 2kth moment of the Dedekind zeta function of a Galois extension is given by the product of the two.[8]

## 메타데이터

### Spacy 패턴 목록

• [{'LOWER': 'dedekind'}, {'LOWER': 'zeta'}, {'LOWER': 'function'}]
• [{'LOWER': 'dedekind'}, {'LOWER': "'s"}, {'LOWER': 'zeta'}, {'LOWER': 'function'}]