Chowla-셀베르그 공식
이 항목의 스프링노트 원문주소==
개요==
Epstein 제타함수==
제1종 타원적분==
- 제1종타원적분 K (complete elliptic integral of the first kind) 에서는 다음과 같은 경우에 대하여, 타원적분의 값을 구체적으로 얻었다
\(\frac{K'}{K}(\frac{1}{\sqrt{2}})= 1\) 이면, \(K(\frac{1}{\sqrt{2}})=\frac{1}{4}B(1/4,1/4)=\frac{\Gamma(\frac{1}{4})^2}{4\sqrt{\pi}}=1.8540746773\cdots\)
\(\frac{K'}{K}(\sqrt{2}-1)= \sqrt{2}\) 이면, \(K(\sqrt{2}-1)=\frac{\sqrt{\sqrt{2}+1}}{2^{13/4}}B(\frac{1}{8},\frac{3}{8})=\frac{\sqrt{\sqrt{2}+1}\Gamma(\frac{1}{8})\Gamma(\frac{3}{8})}{2^{13/4}\sqrt{\pi}}\)
\(\frac{K'}{K}\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)= \sqrt{3}\) 이면, \(K\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)=\frac{\Gamma(\frac{1}{3})\Gamma(\frac{1}{6})}{4\sqrt[4]{3}\sqrt{\pi}}=1.5981420\cdots\)
- lemniscate 곡선의 길이와 타원적분
\(4\int_0^1\frac{dx}{\sqrt{1-x^4}}=B(\frac{1}{2},\frac{1}{4})=\frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})}=\frac{\Gamma(1/4)^2}{\sqrt{2\pi}}=5.24\cdots\)
- 제1종타원적분 K (complete elliptic integral of the first kind)
\(\int_0^1\frac{dx}{\sqrt{1-x^3}}=\frac{1}{3}B(\frac{1}{2},\frac{1}{3})=\frac{1}{6}B(\frac{1}{3},\frac{1}{6})\)
\(6\int_{0}^{1} \frac{dx}{\sqrt{1-x^3}}=B(\frac{1}{3},\frac{1}{6})=\frac{\Gamma(\frac{1}{3})\Gamma(\frac{1}{6})}{\Gamma(\frac{1}{2})}=\frac{\Gamma(\frac{1}{3})\Gamma(\frac{1}{6})}{\sqrt{\pi}}=8.413\cdots\)
\(\frac{K'}{K}(\frac{1}{\sqrt{2}})= 1\) 이면, \(K(\frac{1}{\sqrt{2}})=\frac{1}{4}B(1/4,1/4)=\frac{\Gamma(\frac{1}{4})^2}{4\sqrt{\pi}}=1.8540746773\cdots\)
\(\frac{K'}{K}(\sqrt{2}-1)= \sqrt{2}\) 이면, \(K(\sqrt{2}-1)=\frac{\sqrt{\sqrt{2}+1}}{2^{13/4}}B(\frac{1}{8},\frac{3}{8})=\frac{\sqrt{\sqrt{2}+1}\Gamma(\frac{1}{8})\Gamma(\frac{3}{8})}{2^{13/4}\sqrt{\pi}}\)
\(\frac{K'}{K}\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)= \sqrt{3}\) 이면, \(K\left(\frac{\sqrt{6}-\sqrt{2}}{4}\right)=\frac{\Gamma(\frac{1}{3})\Gamma(\frac{1}{6})}{4\sqrt[4]{3}\sqrt{\pi}}=1.5981420\cdots\)
\(4\int_0^1\frac{dx}{\sqrt{1-x^4}}=B(\frac{1}{2},\frac{1}{4})=\frac{\Gamma(\frac{1}{2})\Gamma(\frac{1}{4})}{\Gamma(\frac{3}{4})}=\frac{\Gamma(1/4)^2}{\sqrt{2\pi}}=5.24\cdots\)
\(\int_0^1\frac{dx}{\sqrt{1-x^3}}=\frac{1}{3}B(\frac{1}{2},\frac{1}{3})=\frac{1}{6}B(\frac{1}{3},\frac{1}{6})\)
\(6\int_{0}^{1} \frac{dx}{\sqrt{1-x^3}}=B(\frac{1}{3},\frac{1}{6})=\frac{\Gamma(\frac{1}{3})\Gamma(\frac{1}{6})}{\Gamma(\frac{1}{2})}=\frac{\Gamma(\frac{1}{3})\Gamma(\frac{1}{6})}{\sqrt{\pi}}=8.413\cdots\)
Chowla-셀베르그의 정리==
Chowla-셀베르그의 정리의 특수한 경우==
- 소수 p에 대하여, 복소이차수체 \(K=\mathbb{Q}(\sqrt{-p})\)의 class number 가 1인 경우, 제1종타원적분 K에 대하여 다음이 성립한다.
- \(\frac{K'}{K}(k)=\sqrt{p}\) 를 만족시키는 k를 찾자.
\(\frac{K(k)}{2\pi}=\frac{2^{1/3}(kk')^{-1/6}}{\sqrt{2\pi p}}\{\prod_{m=1}^{|d_K|}\Gamma(\frac{m}{|d_K|})^{\left(\frac{d_K}{m}\right)}\}^{w_{K}}\)
- \(p=3\)인 경우
\(\frac{K}{2\pi}=\frac{2^{2/3}}{\sqrt{6\pi}}(\frac{\Gamma(\frac{1}{3})}{\Gamma(\frac{2}{3})})^{3/2}\)
- \(p=7\)인 경우
\(\frac{K}{2\pi}=\frac{2}{\sqrt{14\pi}}\sqrt{\frac{\Gamma(\frac{1}{7})\Gamma(\frac{2}{7})\Gamma(\frac{4}{7})}{\Gamma(\frac{3}{7})\Gamma(\frac{5}{7})\Gamma(\frac{6}{7})}}\)
\(\frac{K(k)}{2\pi}=\frac{2^{1/3}(kk')^{-1/6}}{\sqrt{2\pi p}}\{\prod_{m=1}^{|d_K|}\Gamma(\frac{m}{|d_K|})^{\left(\frac{d_K}{m}\right)}\}^{w_{K}}\)
\(\frac{K}{2\pi}=\frac{2^{2/3}}{\sqrt{6\pi}}(\frac{\Gamma(\frac{1}{3})}{\Gamma(\frac{2}{3})})^{3/2}\)
\(\frac{K}{2\pi}=\frac{2}{\sqrt{14\pi}}\sqrt{\frac{\Gamma(\frac{1}{7})\Gamma(\frac{2}{7})\Gamma(\frac{4}{7})}{\Gamma(\frac{3}{7})\Gamma(\frac{5}{7})\Gamma(\frac{6}{7})}}\)
역사==
메모==
- p-adic case Gross-Koblitz form
관련된 항목들==
수학용어번역==
사전 형태의 자료==
관련논문==
- The lord of the numbers, Atle Selberg. On his life and mathematics
- Baas, Nils A.; Skau, Christian F. (2008), Bull. Amer. Math. Soc. 45: 617–649,
- Interview with Selberg
- On the periods of abelian integrals and a formula of Chowla and Selberg
- Benedict H. Gross, Inventiones Mathematicae, Volume 45, Number 2 / 1978년 6월
- On Epstein's Zeta-function
- S. Chowla; A. Selberg, J. reine angew. Math. 227, 86-110, 1967
- On Epstein's Zeta Function (I)
- S. Chowla and A. Selberg Proc Natl Acad Sci U S A. 1949 July; 35(7): 371–374
- On Epstein's Zeta Function
- Max F. Deuring, The Annals of Mathematics, Second Series, Vol. 38, No. 3 (Jul., 1937), pp. 585-593
- Baas, Nils A.; Skau, Christian F. (2008), Bull. Amer. Math. Soc. 45: 617–649,
- Interview with Selberg
- Benedict H. Gross, Inventiones Mathematicae, Volume 45, Number 2 / 1978년 6월
- S. Chowla; A. Selberg, J. reine angew. Math. 227, 86-110, 1967
- S. Chowla and A. Selberg Proc Natl Acad Sci U S A. 1949 July; 35(7): 371–374
- Max F. Deuring, The Annals of Mathematics, Second Series, Vol. 38, No. 3 (Jul., 1937), pp. 585-593