복소 이차 수체의 데데킨트 제타함수 special values

개요

• 데데킨트 제타함수의 구체적인 예

 

\(s=1\) 에서의 값

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

 

 

\(s=2\) 에서의 값

• 복소이차수체의 경우\[\zeta_{K}(2)=\frac{\pi^2}{6\sqrt{|d_K|}}\sum_{(a,d_k)=1} (\frac{d_K}{a})D(e^{2\pi ia/|d_k|})\]\[\zeta_{\mathbb{Q}\sqrt{-7}}(2)=\frac{\pi^2}{3\sqrt{7}}(D(e^{2\pi i/7})+D(e^{4\pi i/7})-D(e^{6\pi i/7}))\]
  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