カテゴリー: ゼータ関数

ゼータ関数$\zeta\left(s\right)$は$\Re\left(s\right)>1$で絶対収束

\[ \zeta\left(1-s,a\right)=\frac{\Gamma\left(s\right)}{\left(2\pi\right)^{s}}\left\{ e^{-i\frac{\pi s}{2}}\Li_{s}\left(e^{2\pi ia}\right)+e^{i\frac{\pi s}{2}}\Li_{s}\left(e^{-2\pi ia}\right)\right\} \]

\[ \zeta\left(-n,\alpha\right)=-\frac{1}{n+1}B_{n+1}\left(\alpha\right) \]

\[ \zeta\left(s,\alpha\right)=-\frac{\Gamma\left(1-s\right)}{2\pi i}\int_{C}\frac{\left(-z\right)^{s-1}e^{-\alpha z}}{1-e^{-z}}dz \]

\[ \zeta\left(s,\alpha\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{t^{s-1}e^{-\alpha t}}{1-e^{-t}}dt \]

\[ \frac{\partial^{n}}{\partial z^{n}}\zeta\left(s,z\right)=P\left(-s,n\right)\zeta\left(s+n,z\right) \]

\[ \zeta\left(s,1\right)=\zeta\left(s\right) \]

\[ \zeta\left(s,\alpha\right)=\sum_{k=0}^{\infty}\frac{1}{\left(\alpha+k\right)^{s}} \]

\[ \prod_{k=1}^{\infty}k=\sqrt{2\pi} \]

\[ \zeta'\left(0\right)=-\Log\sqrt{2\pi} \]