디리클레 L-함수와 수학의 상수들

디리클레 베타함수

 

 

• 원주율(파이,π)
  \(L_{d_K}(1)=\frac{2\pi h_K}{w_K \cdot \sqrt{|d_K|}}\)
  라이프니츠 급수
   
  \(1 \,-\, \frac{1}{3} \,+\, \frac{1}{5} \,-\, \frac{1}{7} \,+\, \frac{1}{9} \,-\, \cdots \;=\; \frac{\pi}{4}\)
  
• 카탈란 상수
  \(G = \beta(2) = \sum_{n=0}^{\infty} \frac{(-1)^{n}}{(2n+1)^2} = \frac{1}{1^2} - \frac{1}{3^2} + \frac{1}{5^2} - \frac{1}{7^2} + \cdots \!=0.915965594\cdots\)
  

• 오일러상수, 감마
  \(L_{d_K}'(1)=\frac{2\pi h_K(\gamma+\ln 2\pi)}{w_K \cdot \sqrt{|d_K|}}-\frac{\pi}{\sqrt{|d_K|}}\sum_{(a,d_K)=1}\chi(a)\log\Gamma (\frac{a}{|d_K|})\)
  \(L_{-4}'(1)=\frac{\pi}{4}(\gamma+\ln 2\pi)-\frac{\pi}{2}\ln(\frac{\Gamma(1/4)}{\Gamma(3/4)})\)