디리클레 L-함수의 미분
개요
- \(d_K\)를 판별식으로 갖는 복소이차수체 \(K\)에 대하여, 디리클레 L-함수는 다음을 만족시킴
\[L_{d_K}'(1)=\frac{2\pi h_K(\gamma+\ln 2\pi)}{w_K \cdot \sqrt{|d_K|}}-\frac{\pi}{\sqrt{|d_K|}}\sum_{(a,d_K)=1}\chi(a)\log\Gamma (\frac{a}{|d_K|})\]
예1
- 정리
수체 \(K=\mathbb{Q}(i)\)에 대하여 다음이 성립한다 \[\beta'(1)=L_{-4}'(1)=\frac{\pi}{4}(\gamma+\ln 2\pi)-\frac{\pi}{2}\ln(\frac{\Gamma(1/4)}{\Gamma(3/4)})\] 여기서 \(\beta\)는 디리클레 베타함수
- 증명
\(\chi\)가 \(\chi(1)=1,\chi(3)=-1\)인 주기가 4인 디리클레 캐릭터라고 하면, \(L_{-4}\)는 다음과 같이 쓸 수 있다. \[L(s):=L_{-4}(s) =\sum_{n\geq 1}\frac{\chi(n)}{n^s}\] 이제 \(L'(1)\) 의 값을 구하자. 후르비츠 제타함수(Hurwitz zeta function)를 이용한 \(L\)-함수의 표현 \[L(s)=4^{-s}\{\zeta(s,1/4)-\zeta(s,3/4)\}\]과 후르비츠 제타함수(Hurwitz zeta function) 의 에르미트 표현 \[\frac{\partial }{\partial s}\zeta(s,a)|_{s=0} =\log \frac{\Gamma(a)}{\sqrt{2\pi}}\] 을 사용하면, 다음을 얻는다. \[L'(s)=4^{-s}\{\zeta(s,1/4)-\zeta(s,3/4)\}(-\log 4)+4^{-s}\{\zeta'(s,1/4)-\zeta'(s,3/4)\}\] 따라서 \[L'(0)=\{\zeta(0,1/4)-\zeta(0,3/4)\}(-\log 4)+\{\zeta'(0,1/4)-\zeta'(0,3/4)\}=-L(0)\log4+\log\frac{\Gamma(1/4)}{\Gamma(3/4)}\] 다음의 함수 \[\Lambda(s)=(\frac{\pi}{4})^{-{(s+1)}/{2}}\Gamma(\frac{s+1}{2})L(s)\] 가 만족시키는 함수방정식 \[\Lambda(s)=\Lambda(1-s)\] 을 사용하자. \(L(0)=\frac{1}{2}\) 을 쉽게 얻을 수 있다.
한편 다이감마 함수(digamma function) 의 값 \[\psi\left(\frac{1}{2}\right) = -2\ln{2} - \gamma\]에서 \(\Gamma'(1/2)=-\sqrt{\pi}(2\ln2+\gamma)\)를 얻고, 이로부터 \[L'(1)=\frac{\pi}{4}\gamma+\frac{\pi}{2}\ln(\frac{\Gamma(3/4)}{\Gamma(1/4)}\sqrt{2\pi})\] 를 얻는다.
따라서 \[L'(1)- \frac{\pi}{4}\gamma=\frac{\pi}{2}\ln(\frac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})}\sqrt{2\pi})\]
예2
- 수체 \(K=\mathbb{Q}(\omega)\), \(\omega^2+\omega+1=0\)에 대하여 다음이 성립한다
\[L_{-3}'(1)=\frac{\pi}{3\sqrt{3}}(\gamma+\ln 2\pi)-\frac{\pi}{\sqrt{3}}\ln(\frac{\Gamma(1/3)}{\Gamma(2/3)})\]
테이블
- 아래의 표에서 \(\epsilon=10^{-6}\)
\[ \begin{array}{c|c|c|c|c} K & L_K'(1-\epsilon ) & L_K'(\epsilon +1) & \sum _{a=1}^{\left|d_K\right|-1} \chi (a) \log \left(\Gamma \left(\frac{a}{\left|d_K\right|}\right)\right) & \sum _{a=1}^{\left|d_K\right|-1} \chi (a) \log \left(\Gamma \left(\frac{a}{\left|d_K\right|}\right)\right) \\ \hline \mathbb{Q}(i) & 0.1929015 & 0.1929012 & 0.1929013 & \frac{1}{4} \pi (\gamma +\log (2 \pi ))-\frac{1}{2} \pi \log \left(\frac{\Gamma \left(\frac{1}{4}\right)}{\Gamma \left(\frac{3}{4}\right)}\right) \\ \mathbb{Q}\left(i \sqrt{2}\right) & -0.02300448 & -0.02300470 & -0.02300459 & \frac{\pi (\gamma +\log (2 \pi ))}{2 \sqrt{2}}-\frac{\pi \log \left(\frac{\Gamma \left(\frac{1}{8}\right) \Gamma \left(\frac{3}{8}\right)}{\Gamma \left(\frac{5}{8}\right) \Gamma \left(\frac{7}{8}\right)}\right)}{2 \sqrt{2}} \\ \mathbb{Q}\left(i \sqrt{3}\right) & 0.2226631 & 0.2226629 & 0.2226630 & \frac{\pi (\gamma +\log (2 \pi ))}{3 \sqrt{3}}-\frac{\pi \log \left(\frac{\Gamma \left(\frac{1}{3}\right)}{\Gamma \left(\frac{2}{3}\right)}\right)}{\sqrt{3}} \\ \mathbb{Q}\left(i \sqrt{5}\right) & -0.4460964 & -0.4460956 & -0.4460960 & \frac{\pi (\gamma +\log (2 \pi ))}{\sqrt{5}}-\frac{\pi \log \left(\frac{\Gamma \left(\frac{1}{20}\right) \Gamma \left(\frac{3}{20}\right) \Gamma \left(\frac{7}{20}\right) \Gamma \left(\frac{9}{20}\right)}{\Gamma \left(\frac{11}{20}\right) \Gamma \left(\frac{13}{20}\right) \Gamma \left(\frac{17}{20}\right) \Gamma \left(\frac{19}{20}\right)}\right)}{2 \sqrt{5}} \\ \mathbb{Q}\left(i \sqrt{6}\right) & -0.4226378 & -0.4226366 & -0.4226372 & \frac{\pi (\gamma +\log (2 \pi ))}{\sqrt{6}}-\frac{\pi \log \left(\frac{\Gamma \left(\frac{1}{24}\right) \Gamma \left(\frac{5}{24}\right) \Gamma \left(\frac{7}{24}\right) \Gamma \left(\frac{11}{24}\right)}{\Gamma \left(\frac{13}{24}\right) \Gamma \left(\frac{17}{24}\right) \Gamma \left(\frac{19}{24}\right) \Gamma \left(\frac{23}{24}\right)}\right)}{2 \sqrt{6}} \\ \mathbb{Q}\left(i \sqrt{7}\right) & 0.01856617 & 0.01856579 & 0.01856598 & \frac{\pi (\gamma +\log (2 \pi ))}{\sqrt{7}}-\frac{\pi \log \left(\frac{\Gamma \left(\frac{1}{7}\right) \Gamma \left(\frac{2}{7}\right) \Gamma \left(\frac{4}{7}\right)}{\Gamma \left(\frac{3}{7}\right) \Gamma \left(\frac{5}{7}\right) \Gamma \left(\frac{6}{7}\right)}\right)}{\sqrt{7}} \end{array} \]
메모
- 리만제타함수는 다음을 만족한다
\[\zeta'(0)=-\log{\sqrt{2\pi}}\]
관련된 항목들
매스매티카 파일 및 계산 리소스
관련논문
- Yang, T. (2010). The Chowla-Selberg formula and the Colmez conjecture. Canad. J. Math, 62(2), 456-472. http://www.math.wisc.edu/~thyang/ColmezConjectureFinal2010.pdf
- Anderson, G. W. (1982). Logarithmic derivatives of Dirichlet \( L \)-functions and the periods of abelian varieties. Compositio Mathematica, 45(3), 315-332.