데데킨트 제타함수
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개요
- 수체 \(K\)에 대하여, 데데킨트 제타함수는 다음과 같이 정의됨
\[\zeta_{K}(s):=\sum_{I \text{:ideals}}\frac{1}{N(I)^s}\]
- 예
- \(K=\mathbb{Q}\) 인 경우, 리만제타함수를 얻음
- 전체 복소평면으로 해석적확장(analytic continuation) 되며, \(s=1\) 에서 simple pole을 가진다
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
- \(s=1\) 에서의 유수(residue)는 디리클레 유수 (class number) 공식으로 주어진다
\[ \lim_{s\to 1} (s-1)\zeta_K(s)=\frac{2^{r_1}\cdot(2\pi)^{r_2}\cdot h_K\cdot R_K}{w_K \cdot \sqrt{|D_K|}}\]
- \(s=0\) 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\[L(\chi,s) =\sum_{\mathfrak{a} \text{:ideals}}\frac{\chi(\mathfrak{a})}{N(\mathfrak{a})^s} = \sum_{A\in C_K}{\chi(A)}\zeta_K(s,A)\]
예
special values
클링겐-지겔 (Klingen-Siegel) 정리
- 클링겐-지겔 (Klingen-Siegel) 정리
- F : totally real 수체
- \([F: \mathbb{Q}]=n\)
- \(m>0\)일 때, 다음을 만족하는 적당한 유리수 \(r(m)\in \mathbb{Q}\)가 존재한다
\[\zeta_{F}(2m)=r(m)\frac{\pi^{2mn}}{\sqrt{|d_{F}|}}\]
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
역사
메모
- http://www.umpa.ens-lyon.fr/~brunault/recherche/parma.pdf
- http://mathoverflow.net/questions/87873/dedekind-zeta-function-behaviour-at-1
관련된 항목들
계산 리소스
- https://docs.google.com/file/d/0B8XXo8Tve1cxcXFHOEFSMHc1bUk/edit
- Tables of Values of Dedekind Zeta Functions
사전 형태의 자료
리뷰, 에세이, 강의노트
- H. M. Stark, "Galois theory, algebraic number theory and zeta functions" ,in \ From number theory to physics", ed. M. Walschmidt, P. Moussa, J.-M. Luck, C. Itzykson Springer
- H. M. Stark, The analytic theory of algebraic numbers http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183537391
- Matilde N. Lalin, Hyperbolic volumes and zeta values An introduction
관련논문
- Zagier, Don. ‘Hyperbolic Manifolds and Special Values of Dedekind Zeta-Functions’. Inventiones Mathematicae 83, no. 2 (1 June 1986): 285–301. doi:10.1007/BF01388964.
- D. Zagier, Polylogarithms, Dedekind zeta functions and the algebraic K-theory of fields
- Borel, A. ‘Commensurability Classes and Volumes of Hyperbolic 3-Manifolds’. Annali Della Scuola Normale Superiore Di Pisa - Classe Di Scienze 8, no. 1 (1981): 1–33.
노트
말뭉치
- 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]
- 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]
- Just like the Riemann zeta function, each Dedekind zeta function possesses a functional equation.[3]
- The nontrivial zeros of the Dedekind zeta function of any algebraic number eld lie on the critical line: Re(s) = 1/2.[4]
- Theorem Let X be a group of Dirichlet characters, K the associated eld, and K (s) the Dedekind zeta function of K .[4]
- From there, we discuss algebraic number elds and introduce the tools needed to dene the Dedekind zeta function.[5]
- 1 2 FRIMPONG A. BAIDOO necessary for providing context to the Dedekind zeta function.[5]
- 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]
- I was trying to learn a little about the Dedekind zeta function.[6]
- 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]
- We study analytic aspects of the Dedekind zeta function of a Galois extension.[8]
- 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]
- In the second part, we derive a hybrid Euler-Hadamard product for the Dedekind zeta function of an arbitrary number eld.[8]
- 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]
소스
- ↑ Dedekind zeta function
- ↑ Dedekind zeta function in nLab
- ↑ Dedekind zeta function
- ↑ 4.0 4.1 Introduction to l-functions:
- ↑ 5.0 5.1 5.2 Dirichlet l-functions and dedekind ζ-functions
- ↑ Relation between the Dedekind Zeta Function and Quadratic Reciprocity
- ↑ Mean values connected with the Dedekind zeta function
- ↑ 8.0 8.1 8.2 8.3 Moments of the dedekind zeta function
메타데이터
위키데이터
- ID : Q1182160
Spacy 패턴 목록
- [{'LOWER': 'dedekind'}, {'LOWER': 'zeta'}, {'LOWER': 'function'}]
- [{'LOWER': 'dedekind'}, {'LOWER': "'s"}, {'LOWER': 'zeta'}, {'LOWER': 'function'}]