"복소 이차 수체의 데데킨트 제타함수 special values"의 두 판 사이의 차이

2020년 11월 13일 (금) 07:27 기준 최신판

개요

복소이차수체의 데데킨트 제타함수

\(s=1\) 에서의 값

이차 수체에 대한 디리클레 class number 공식
\(K=\mathbb{Q}(\sqrt{-q})\)

\(q \equiv 3 \pmod{4}\)

\(K=\mathbb{Q}(\sqrt{-q})\), \(q \geq 7\) , \(q \equiv 3 \pmod{4}\) 인 경우
\(d_K=-q\)
\(\chi(a)=\left(\frac{a}{q}\right)\)
\(\chi(-1)=-1\), \(\tau(\chi)=i\sqrt{q}\)
다음이 성립한다

\[L(1,\chi)= \frac{- \pi\sqrt{q}}{q^2}\sum_{a=1}^{q-1}\left(\frac{a}{q}\right) a=\frac{\pi h_K}{\sqrt{q}}\] \[h_K=-\sum_{a=1}^{q-1}\left(\frac{a}{q}\right)\frac{a}{q}\]

\(q \equiv 1 \pmod{4}\)

\(K=\mathbb{Q}(\sqrt{-q})\) , \(q \geq 5\) , \(q \equiv 1 \pmod{4}\) 인 경우
\(d_K=-4q\), \(\chi(-1)=-1\), \(\tau(\chi)=2i\sqrt{q}\)
다음이 성립한다

\[L(1,\chi)= -\frac{ \pi\sqrt{q}}{8q^2}\sum_{(a,4q)=1}\chi(a) a=\frac{\pi h_K}{2\sqrt{q}}\] \[h_K=-\frac{1}{4}\sum_{(a,4q)=1}\left(\frac{a}{q}\right)\frac{a}{q}\]

\(s=2\) 에서의 값

복소이차수체 \(K\)에 대하여 다음이 성립한다

\[\zeta_{K}(2)=\frac{\pi^2}{6\sqrt{|d_K|}}\sum_{(a,d_k)=1} (\frac{d_K}{a})D(e^{2\pi ia/|d_k|})\]

예를 들어, \(K=\mathbb{Q}\sqrt{-7}\)에 대하여, 다음이 성립한다

\[\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))\] 여기서 \(D(z)\)는 블로흐-비그너 다이로그(Bloch-Wigner dilogarithm)

예

\[\zeta_{\mathbb{Q}\sqrt{-1}}(2)=1.50670301\] \[\zeta_{\mathbb{Q}\sqrt{-2}}(2)=1.75141751\cdots\] \[\zeta_{\mathbb{Q}\sqrt{-3}}(2)=\frac{\pi^2}{6\sqrt{3}}(D(e^{2\pi i/3})-D(e^{4\pi i/3}))=\frac{\pi^2}{3\sqrt{3}}D(e^{2\pi i/3})=1.285190955484149\cdots\] \[\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=1.89484145\] \[\zeta_{\mathbb{Q}\sqrt{-11}}(2)=1.49613186\]

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

figure eight knot complement

\(V=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots\)
\(\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=\frac{\pi^2}{3\sqrt{3}}D(e^{\frac{2\pi i}{3}})\)
\(L_{-3}(2)=\frac{2}{\sqrt{3}}D(e^{\frac{2\pi i}{3}})\)
2.02988321281930725

\[V(4_{1})=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots\]

매듭이론 (knot theory)

메모

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

매스매티카 파일 및 계산 리소스

https://docs.google.com/file/d/0B8XXo8Tve1cxc00tSVRvMXluOWM/edit

@@ 1번째 줄: / 1번째 줄: @@
 ==개요==
-* [[데데킨트 제타함수]]의 구체적인 예
+* 복소이차수체의 [[데데킨트 제타함수]]
+==<math>s=1</math> 에서의 값==
+* [[이차 수체에 대한 디리클레 class number 공식 |이차 수체에 대한 디리클레 class number 공식]]
+* <math>K=\mathbb{Q}(\sqrt{-q})</math>
-==<math>s=1</math> 에서의 값==
-* [[이차 수체에 대한 디리클레 class number 공식 |이차 수체에 대한 디리클레 class number 공식]]<br>
+===<math>q \equiv 3 \pmod{4}</math>===
-*  복소이차수체의 경우:<math>K=\mathbb{Q}(\sqrt{-q})</math>, <math>q \geq 7</math> , <math>q \equiv 3 \pmod{4}</math> 인 경우:<math>d_K=-q</math>:<math>\chi(a)=\left(\frac{a}{q}\right)</math>:<math>\chi(-1)=-1</math>, <math>\tau(\chi)=i\sqrt{q}</math>:<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>:<math>h_K=-\sum_{a=1}^{q-1}\left(\frac{a}{q}\right)\frac{a}{q}</math><br>  :<math>K=\mathbb{Q}(\sqrt{-q})</math>  , <math>q \geq 5</math> ,  <math>q \equiv 1 \pmod{4}</math> 인 경우:<math>d_K=-4q</math>:<math>\chi(-1)=-1</math>, <math>\tau(\chi)=2i\sqrt{q}</math>:<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>:<math>h_K=-\frac{1}{4}\sum_{(a,4q)=1}\left(\frac{a}{q}\right)\frac{a}{q}</math><br>
+* <math>K=\mathbb{Q}(\sqrt{-q})</math>, <math>q \geq 7</math> , <math>q \equiv 3 \pmod{4}</math> 인 경우
+* <math>d_K=-q</math>
+* <math>\chi(a)=\left(\frac{a}{q}\right)</math>
+* <math>\chi(-1)=-1</math>, <math>\tau(\chi)=i\sqrt{q}</math>
+* 다음이 성립한다
+:<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>
+:<math>h_K=-\sum_{a=1}^{q-1}\left(\frac{a}{q}\right)\frac{a}{q}</math>
+===<math>q \equiv 1 \pmod{4}</math>===
+* <math>K=\mathbb{Q}(\sqrt{-q})</math>  , <math>q \geq 5</math> ,  <math>q \equiv 1 \pmod{4}</math> 인 경우
+* <math>d_K=-4q</math>, <math>\chi(-1)=-1</math>, <math>\tau(\chi)=2i\sqrt{q}</math>
+* 다음이 성립한다
+:<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>
+:<math>h_K=-\frac{1}{4}\sum_{(a,4q)=1}\left(\frac{a}{q}\right)\frac{a}{q}</math>
-==<math>s=2</math> 에서의 값==
-*  복소이차수체의 경우:<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>:<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>
+==<math>s=2</math> 에서의 값==
-*  예:<math>\zeta_{\mathbb{Q}\sqrt{-1}}(2)=1.50670301</math>:<math>\zeta_{\mathbb{Q}\sqrt{-2}}(2)=1.75141751\cdots</math>:<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>:<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>:<math>\zeta_{\mathbb{Q}\sqrt{-11}}(2)=1.49613186</math><br>
+*  복소이차수체 <math>K</math>에 대하여 다음이 성립한다
+:<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>
+* 예를 들어, <math>K=\mathbb{Q}\sqrt{-7}</math>에 대하여, 다음이 성립한다
+:<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> 여기서 <math>D(z)</math>는 [[블로흐-비그너 다이로그(Bloch-Wigner dilogarithm)]]
+* 예
+:<math>\zeta_{\mathbb{Q}\sqrt{-1}}(2)=1.50670301</math>
+:<math>\zeta_{\mathbb{Q}\sqrt{-2}}(2)=1.75141751\cdots</math>
+:<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>
+:<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>
+:<math>\zeta_{\mathbb{Q}\sqrt{-11}}(2)=1.49613186</math>
-* 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>
+* 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
+===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
+:<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>
+* [[매듭이론 (knot theory)]]
-==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:<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=
-* [[수학사 연표]]
 ==메모==
-* <math>s=1</math> 에서의 <math>L_{d_K}'(1)</math>의 값:<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>
+* <math>s=1</math> 에서의 <math>L_{d_K}'(1)</math>의 값:<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>
-* [[L-함수의 미분]] 항목 참조<br>
+* [[L-함수의 미분]] 항목 참조
-* Math Overflow http://mathoverflow.net/search?q=
 ==관련된 항목들==
+* [[이차 수체(quadratic number fields) 의 정수론]]
-==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==
+* https://docs.google.com/file/d/0B8XXo8Tve1cxc00tSVRvMXluOWM/edit
-*
-* 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==
+[[분류:정수론]]