"Zeta value at 2"의 두 판 사이의 차이

수학노트
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(사용자 2명의 중간 판 14개는 보이지 않습니다)
1번째 줄: 1번째 줄:
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">introduction</h5>
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==개요==
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* {{수학노트|url=복소_이차_수체의_데데킨트_제타함수_special_values}} 항목으로 옮겨짐
  
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*  복소이차수체의 [http://pythagoras0.springnote.com/pages/4533335 데데킨트 제타함수]<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>
 
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*  Note that<br>
 
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**  the Clausen function and the Bloch-Wigner dilogarithms are same if <math>z=e^{i\theta}</math><br><math>\operatorname{Cl}_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}</math><br><math>D(z)=\text{Im}(\operatorname{Li}_2(z))+\log|z|\arg(1-z)</math><br>
 
  
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==related items==
  
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* [[Volume of hyperbolic 3-manifolds]]
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* [[Kashaev's volume conjecture]]
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<h5 style="line-height: 2em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">a few examples</h5>
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* [[Central charge and dilogarithm]]
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* 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>
 
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<math>\zeta_{\mathbb{Q}\sqrt{-1}}(2)=1.50670301</math>
 
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<math>\zeta_{\mathbb{Q}\sqrt{-2}}(2)=1.75141751\cdots</math>
 
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<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>
 
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<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>
 
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<math>\zeta_{\mathbb{Q}\sqrt{-11}}(2)=1.49613186</math>
 
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#  Cl[x_] := Im[PolyLog[2, Exp[I*x]]]<br> disc[n_] := NumberFieldDiscriminant[Sqrt[-n]]<br> L2[n_] :=<br>  1/Sqrt[Abs[disc[n]]]*<br>   Sum[JacobiSymbol[disc[n], k] Cl[2 Pi*k/Abs[disc[n]]], {k, 1,<br>     Abs[disc[n]] - 1}]<br> Zeta2[n_] := L2[n]*Pi^2/6<br> Zeta2[1]<br>
 
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<h5 style="line-height: 2em; margin: 0px;">figure eight knot complement</h5>
 
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<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>
 
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<math>\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=\frac{\pi^2}{3\sqrt{3}}D(e^{\frac{2\pi i}{3}})</math>
 
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<math>L_{-3}(2)=\frac{2}{\sqrt{3}}D(e^{\frac{2\pi  i}{3}})</math>
 
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*  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> where D is [http://pythagoras0.springnote.com/pages/4633853 Bloch-Wigner dilogarithm].<br>
 
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*  what is <math>\zeta_{\mathbb{Q}(\sqrt{-3})}(2)</math>? numrically 1.285190955484149<br>
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">history</h5>
 
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">related items</h5>
 
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* [http://pythagoras0.springnote.com/pages/2005116 이차 수체에 대한 디리클레 class number 공식]
 
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* [[4593563|Dirichlet class number formula]]
 
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* [[volume of hyperbolic threefolds and L-values]]
 
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* [[Kashaev's volume conjecture|Kashaev's volume Conjecture]]
 
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* [[central charge, L-values and dilogarithm|central charge of CFT, L-values and dilogarithm]]
 
 
* [[Gieseking's constant]]
 
* [[Gieseking's constant]]
  
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[[분류:개인노트]]
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[[분류:Number theory and physics]]
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[[분류:L-functions and L-values]]
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[[분류:migrate]]
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* http://ko.wikipedia.org/wiki/
 
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* http://en.wikipedia.org/wiki/
 
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* Princeton companion to mathematics([[2910610/attachments/2250873|Companion_to_Mathematics.pdf]])
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">books</h5>
 
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* [[2010년 books and articles]]<br>
 
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* http://gigapedia.info/1/
 
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* http://gigapedia.info/1/
 
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">articles</h5>
 
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* [[2010년 books and articles|논문정 리]]
 
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* http://www.ams.org/mathscinet
 
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
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* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
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* http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
 
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* http://dx.doi.org/
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">question and answers(Math Overflow)</h5>
 
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">blogs</h5>
 
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*  구글 블로그 검색<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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** http://blogsearch.google.com/blogsearch?q=
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">experts on the field</h5>
 
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* http://arxiv.org/
 
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<h5 style="line-height: 3.428em; margin: 0px; color: rgb(34, 61, 103); font-family: 'malgun gothic',dotum,gulim,sans-serif; font-size: 1.166em; background-position: 0px 100%;">links</h5>
 
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* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier]
 
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* [http://pythagoras0.springnote.com/pages/1947378 수식표 현 안내]
 
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* [http://www.research.att.com/%7Enjas/sequences/index.html The On-Line Encyclopedia of Integer Sequences]
 
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* http://functions.wolfram.com/
 
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2020년 11월 16일 (월) 07:00 기준 최신판

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