"데데킨트 제타함수"의 두 판 사이의 차이

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(같은 사용자의 중간 판 5개는 보이지 않습니다)
53번째 줄: 53번째 줄:
 
* F : totally real 수체
 
* F : totally real 수체
 
* <math>[F: \mathbb{Q}]=n</math>
 
* <math>[F: \mathbb{Q}]=n</math>
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* $m>0$일 때, 다음을 만족하는 적당한 유리수 <math>r(m)\in \mathbb{Q}</math>가 존재한다
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* <math>m>0</math>일 때, 다음을 만족하는 적당한 유리수 <math>r(m)\in \mathbb{Q}</math>가 존재한다
 
:<math>\zeta_{F}(2m)=r(m)\frac{\pi^{2mn}}{\sqrt{|d_{F}|}}</math>
 
:<math>\zeta_{F}(2m)=r(m)\frac{\pi^{2mn}}{\sqrt{|d_{F}|}}</math>
 
* http://planetmath.org/SiegelKlingenTheorem.html
 
* http://planetmath.org/SiegelKlingenTheorem.html
59번째 줄: 59번째 줄:
 
===Zagier, Bloch, Suslin===
 
===Zagier, Bloch, Suslin===
 
* <math>[K : \mathbb{Q}] = r_1 + 2r_2</math>일 때,
 
* <math>[K : \mathbb{Q}] = r_1 + 2r_2</math>일 때,
−
:<math>\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}</math> 여기서 <math>\xi_i,(i=1,\cdots, r_2)</math> 는 Bloch group <math>B(K)\otimes \mathbb{Q}</math>의 $\mathbb{Q}$-basis D는 [[블로흐-비그너 다이로그(Bloch-Wigner dilogarithm)]] 함수이며, <math>a\sim_{\mathbb{Q^{\times}}} b</math> 는 <math>a/b\in\mathbb{Q}</math> 를 의미함
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:<math>\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}</math> 여기서 <math>\xi_i,(i=1,\cdots, r_2)</math> 는 Bloch group <math>B(K)\otimes \mathbb{Q}</math>의 <math>\mathbb{Q}</math>-basis D는 [[블로흐-비그너 다이로그(Bloch-Wigner dilogarithm)]] 함수이며, <math>a\sim_{\mathbb{Q^{\times}}} b</math> 는 <math>a/b\in\mathbb{Q}</math> 를 의미함
  
 
   
 
   
119번째 줄: 119번째 줄:
 
   
 
   
 
[[분류:정수론]]
 
[[분류:정수론]]
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== 노트 ==
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===말뭉치===
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# 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.<ref name="ref_ca3cd66b">[https://en.wikipedia.org/wiki/Dedekind_zeta_function Dedekind zeta function]</ref>
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# 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).<ref name="ref_8e168495">[https://ncatlab.org/nlab/show/Dedekind+zeta+function Dedekind zeta function in nLab]</ref>
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# Just like the Riemann zeta function, each Dedekind zeta function possesses a functional equation.<ref name="ref_96a08161">[https://www.theochem.ru.nl/~pwormer/Knowino/knowino.org/wiki/Dedekind_zeta_function.html Dedekind zeta function]</ref>
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# The nontrivial zeros of the Dedekind zeta function of any algebraic number eld lie on the critical line: Re(s) = 1/2.<ref name="ref_e51d5cc2">[http://archive.schools.cimpa.info/archivesecoles/20171023105958/PV-lecture2.pdf Introduction to l-functions:]</ref>
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# Theorem Let X be a group of Dirichlet characters, K the associated eld, and K (s) the Dedekind zeta function of K .<ref name="ref_e51d5cc2" />
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# From there, we discuss algebraic number elds and introduce the tools needed to dene the Dedekind zeta function.<ref name="ref_d04421dc">[https://math.uchicago.edu/~may/REU2016/REUPapers/Baidoo.pdf Dirichlet l-functions and dedekind ζ-functions]</ref>
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# 1 2 FRIMPONG A. BAIDOO necessary for providing context to the Dedekind zeta function.<ref name="ref_d04421dc" />
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# 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.<ref name="ref_d04421dc" />
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# I was trying to learn a little about the Dedekind zeta function.<ref name="ref_c73b2fb0">[https://math.stackexchange.com/questions/33006/relation-between-the-dedekind-zeta-function-and-quadratic-reciprocity Relation between the Dedekind Zeta Function and Quadratic Reciprocity]</ref>
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# 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.<ref name="ref_2c7bf667">[https://link.springer.com/article/10.1007/s10958-008-0126-9 Mean values connected with the Dedekind zeta function]</ref>
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# We study analytic aspects of the Dedekind zeta function of a Galois extension.<ref name="ref_2aacc359">[https://core.ac.uk/download/pdf/18451908.pdf Moments of the dedekind zeta function]</ref>
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# 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).<ref name="ref_2aacc359" />
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# In the second part, we derive a hybrid Euler-Hadamard product for the Dedekind zeta function of an arbitrary number eld.<ref name="ref_2aacc359" />
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# We then conjecture that the 2kth moment of the Dedekind zeta function of a Galois extension is given by the product of the two.<ref name="ref_2aacc359" />
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===소스===
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<references />
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== 메타데이터 ==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q1182160 Q1182160]
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===Spacy 패턴 목록===
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* [{'LOWER': 'dedekind'}, {'LOWER': 'zeta'}, {'LOWER': 'function'}]
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* [{'LOWER': 'dedekind'}, {'LOWER': "'s"}, {'LOWER': 'zeta'}, {'LOWER': 'function'}]

2021년 2월 26일 (금) 01:41 기준 최신판

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\[L(\chi,s) =\sum_{\mathfrak{a} \text{:ideals}}\frac{\chi(\mathfrak{a})}{N(\mathfrak{a})^s} = \sum_{A\in C_K}{\chi(A)}\zeta_K(s,A)\]



예



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



역사



메모



관련된 항목들


계산 리소스


사전 형태의 자료



리뷰, 에세이, 강의노트



관련논문

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]

소스

  1. ↑ Dedekind zeta function
  2. ↑ Dedekind zeta function in nLab
  3. ↑ Dedekind zeta function
  4. ↑ 4.0 4.1 Introduction to l-functions:
  5. ↑ 5.0 5.1 5.2 Dirichlet l-functions and dedekind ζ-functions
  6. ↑ Relation between the Dedekind Zeta Function and Quadratic Reciprocity
  7. ↑ Mean values connected with the Dedekind zeta function
  8. ↑ 8.0 8.1 8.2 8.3 Moments of the dedekind zeta function

메타데이터

위키데이터

Spacy 패턴 목록

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