Zeta value at 2

introduction

•  

• 복소이차수체의 데데킨트 제타함수
  \(\zeta_{K}(2)=\frac{\pi^2}{6\sqrt{|d_K|}}\sum_{(a,d_k)=1} (\frac{d_K}{a})D(e^{2\pi ia/|d_k|})\)
  
• Note that
  
  • the Clausen function and the Bloch-Wigner dilogarithms are same if \(z=e^{i\theta}\)
    \(\operatorname{Cl}_2(\theta)=-\int_0^{\theta} \ln |2\sin \frac{t}{2}| \,dt=\sum_{n=1}^{\infty}\frac{\sin (n\theta)}{n^2}\)
    \(D(z)=\text{Im}(\operatorname{Li}_2(z))+\log|z|\arg(1-z)\)
    

 

 

a few examples

• 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
  

\(\zeta_{\mathbb{Q}\sqrt{-1}}(2)=1.50670301\)

\(\zeta_{\mathbb{Q}\sqrt{-2}}(2)=1.75141751\cdots\)

\(\zeta_{\mathbb{Q}\sqrt{-3}}(2)=\frac{\pi^2}{6\sqrt{3}}(D(e^{2\pi i/3})-D(e^{4\pi i/3}))=\frac{\pi^2}{3\sqrt{3}}D(e^{2\pi i/3})=1.285190955484149\cdots\)

\(\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))=1.89484145\)

\(\zeta_{\mathbb{Q}\sqrt{-11}}(2)=1.49613186\)

1. Cl[x_] := Im[PolyLog[2, Exp[I*x]]]
   disc[n_] := NumberFieldDiscriminant[Sqrt[-n]]
   L2[n_] :=
    1/Sqrt[Abs[disc[n]]]*
     Sum[JacobiSymbol[disc[n], k] Cl[2 Pi*k/Abs[disc[n]]], {k, 1,
       Abs[disc[n]] - 1}]
   Zeta2[n_] := L2[n]*Pi^2/6
   Zeta2[1]
   

 

 

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
  \(V(4_{1})=\frac{9\sqrt{3}}{\pi^2}\zeta_{\mathbb{Q}(\sqrt{-3})}(2)=3D(e^{\frac{2i\pi}{3}})=2D(e^{\frac{i\pi}{3}})=2.029883212819\cdots\)
  where D is Bloch-Wigner dilogarithm.
  
• what is \(\zeta_{\mathbb{Q}(\sqrt{-3})}(2)\)? numrically 1.285190955484149
  

 

 

history

• http://www.google.com/search?hl=en&tbs=tl:1&q=

 

 

related items

• 이차 수체에 대한 디리클레 class number 공식
• Dirichlet class number formula
• volume of hyperbolic threefolds and L-values
• Kashaev's volume Conjecture
• central charge of CFT, L-values and dilogarithm
• Gieseking's constant

 

 

encyclopedia

• http://ko.wikipedia.org/wiki/
• http://en.wikipedia.org/wiki/
• Princeton companion to mathematics(Companion_to_Mathematics.pdf)

 

 

books

 

• 2010년 books and articles
  
• http://gigapedia.info/1/
• http://gigapedia.info/1/
• http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=

 

 

articles

 

• 논문정 리
• http://www.ams.org/mathscinet
• http://www.zentralblatt-math.org/zmath/en/
• http://pythagoras0.springnote.com/
• http://math.berkeley.edu/~reb/papers/index.html
• http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
• 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=
• http://dx.doi.org/

 

 

question and answers(Math Overflow)

• http://mathoverflow.net/search?q=
• http://mathoverflow.net/search?q=

 

 

blogs

• 구글 블로그 검색
  
  • http://blogsearch.google.com/blogsearch?q=
  • http://blogsearch.google.com/blogsearch?q=

 

 

experts on the field

• http://arxiv.org/

 

 

links

• Detexify2 - LaTeX symbol classifier
• 수식표 현 안내
• The On-Line Encyclopedia of Integer Sequences
• http://functions.wolfram.com/