복소 이차 수체의 데데킨트 제타함수 special values 문서 원본 보기

==이 항목의 수학노트 원문주소==

* [[복소이차수체의 데데킨트 제테함수]]

 

 

==개요==

 

 

 

==<math>s=1</math> 에서의 값==

* [[이차 수체에 대한 디리클레 class number 공식 |이차 수체에 대한 디리클레 class number 공식]]<br>
*  복소이차수체의 경우<br><math>K=\mathbb{Q}(\sqrt{-q})</math>, <math>q \geq 7</math> , <math>q \equiv 3 \pmod{4}</math> 인 경우<br><math>d_K=-q</math><br><math>\chi(a)=\left(\frac{a}{q}\right)</math><br><math>\chi(-1)=-1</math>, <math>\tau(\chi)=i\sqrt{q}</math><br><math>L(1,\chi)= \frac{- \pi\sqrt{q}}{q^2}\sum_{a=1}^{q-1}\left(\frac{a}{q}\right) a=\frac{\pi h_K}{\sqrt{q}}</math><br><math>h_K=-\sum_{a=1}^{q-1}\left(\frac{a}{q}\right)\frac{a}{q}</math><br>  <br><math>K=\mathbb{Q}(\sqrt{-q})</math>  , <math>q \geq 5</math> ,  <math>q \equiv 1 \pmod{4}</math> 인 경우<br><math>d_K=-4q</math><br><math>\chi(-1)=-1</math>, <math>\tau(\chi)=2i\sqrt{q}</math><br><math>L(1,\chi)= -\frac{ \pi\sqrt{q}}{8q^2}{\sum_{(a,4q)=1}\chi(a) a=\frac{\pi h_K}{2\sqrt{q}}</math><br><math>h_K=-\frac{1}{4}\sum_{(a,4q)=1}\left(\frac{a}{q}\right)\frac{a}{q}</math><br>

 

 

==<math>s=2</math> 에서의 값==

*  복소이차수체의 경우<br><math>\zeta_{K}(2)=\frac{\pi^2}{6\sqrt{|d_K|}}\sum_{(a,d_k)=1} (\frac{d_K}{a})D(e^{2\pi ia/|d_k|})</math><br><math>\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))</math><br> 여기서 <math>D(z)</math>는 [[블로흐-비그너 다이로그(Bloch-Wigner dilogarithm)]]<br>
*  예<br><math>\zeta_{\mathbb{Q}\sqrt{-1}}(2)=1.50670301</math><br><math>\zeta_{\mathbb{Q}\sqrt{-2}}(2)=1.75141751\cdots</math><br><math>\zeta_{\mathbb{Q}\sqrt{-3}}(2)=\frac{\pi^2}{6\sqrt{3}}(D(e^{2\pi i/3})-D(e^{4\pi i/3}))=\frac{\pi^2}{3\sqrt{3}}D(e^{2\pi i/3})=1.285190955484149\cdots</math><br><math>\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=1.89484145</math><br><math>\zeta_{\mathbb{Q}\sqrt{-11}}(2)=1.49613186</math><br>

 

* http://books.google.co.kr/books?id=yrmT56mpw3kC&pg=PA367&dq=smallest+norms+of+prime+ideals&hl=ko&ei=IMRTTIaRGoqWvAP88MUZ&sa=X&oi=book_result&ct=result&resnum=4&ved=0CDgQ6AEwAw#v=onepage&q=smallest%20norms%20of%20prime%20ideals&f=false<br>

 

 

 

==figure eight knot complement==

<math>V=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots</math>

<math>\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=\frac{\pi^2}{3\sqrt{3}}D(e^{\frac{2\pi i}{3}})</math>

<math>L_{-3}(2)=\frac{2}{\sqrt{3}}D(e^{\frac{2\pi  i}{3}})</math>

*  2.02988321281930725<br><math>V(4_{1})=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots</math><br>
* [[매듭이론 (knot theory)]]<br>

 

 

==역사==

 

* http://www.google.com/search?hl=en&tbs=tl:1&q=
* [[수학사연표 (역사)|수학사연표]]

 

 

==메모==

* <math>s=1</math> 에서의 <math>L_{d_K}'(1)</math>의 값<br><math>L_{d_K}'(1)=\frac{2\pi h_K(\gamma+\ln 2\pi)}{w_K \cdot \sqrt{|d_K|}}-\frac{\pi}{\sqrt{|d_K|}}\sum_{(a,d_K)=1}\chi(a)\log\Gamma (\frac{a}{|d_K|})</math><br>
* [[L-함수의 미분]] 항목 참조<br>

* Math Overflow http://mathoverflow.net/search?q=

 

 

==관련된 항목들==

 

 

==수학용어번역==

*  단어사전<br>
** http://translate.google.com/#en|ko|
** http://ko.wiktionary.org/wiki/
* 발음사전 http://www.forvo.com/search/
* [http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=&fstr= 대한수학회 수학 학술 용어집]<br>
** http://mathnet.kaist.ac.kr/mathnet/math_list.php?mode=list&ftype=eng_term&fstr=
* [http://www.kss.or.kr/pds/sec/dic.aspx 한국통계학회 통계학 용어 온라인 대조표]
* [http://www.nktech.net/science/term/term_l.jsp?l_mode=cate&s_code_cd=MA 남·북한수학용어비교]
* [http://kms.or.kr/home/kor/board/bulletin_list_subject.asp?bulletinid=%7BD6048897-56F9-43D7-8BB6-50B362D1243A%7D&boardname=%BC%F6%C7%D0%BF%EB%BE%EE%C5%E4%B7%D0%B9%E6&globalmenu=7&localmenu=4 대한수학회 수학용어한글화 게시판]

 

 

==매스매티카 파일 및 계산 리소스==

*  
* http://www.wolframalpha.com/input/?i=
* http://functions.wolfram.com/
* [http://dlmf.nist.gov/ NIST Digital Library of Mathematical Functions]
* [http://people.math.sfu.ca/%7Ecbm/aands/toc.htm Abramowitz and Stegun Handbook of mathematical functions]
* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
* [http://numbers.computation.free.fr/Constants/constants.html Numbers, constants and computation]
* [https://docs.google.com/open?id=0B8XXo8Tve1cxMWI0NzNjYWUtNmIwZi00YzhkLTkzNzQtMDMwYmVmYmIxNmIw 매스매티카 파일 목록]

 

 

==사전 형태의 자료==

* http://ko.wikipedia.org/wiki/
* http://en.wikipedia.org/wiki/
* [http://eom.springer.de/default.htm The Online Encyclopaedia of Mathematics]
* [http://dlmf.nist.gov NIST Digital Library of Mathematical Functions]
* [http://eqworld.ipmnet.ru/ The World of Mathematical Equations]

 

 

==리뷰논문, 에세이, 강의노트==

 

 

 

==관련논문==

* http://www.jstor.org/action/doBasicSearch?Query=
* http://www.ams.org/mathscinet
* http://dx.doi.org/

 

 

==관련도서==

*  도서내검색<br>
** http://books.google.com/books?q=
** http://book.daum.net/search/contentSearch.do?query=