"Zeta value at 2"의 두 판 사이의 차이

수학노트
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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>
 
*  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>

2012년 4월 18일 (수) 15:57 판

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

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

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

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

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

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

  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

 

 

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