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| Both sides previous revision Previous revision Next revision | Previous revision Next revision Both sides next revision | ||
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riemann_zeta_function [2015/12/27 17:23] nikolaj |
riemann_zeta_function [2016/06/02 10:58] nikolaj |
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| == Functional equation == | == Functional equation == | ||
| Tells you most values: | Tells you most values: | ||
| - | ^ $ \zeta(s) = 2(2\pi)^{s-1}\sin{\left(\pi\,s/2\right)}\,\Gamma(1-s)\,\zeta(1-s)$ ^ | + | ^ $ \zeta(s) = 2\,(2\pi)^{s-1}\sin{\left(\pi\,s/2\right)}\,\Gamma(1-s)\,\zeta(1-s)$ ^ |
| == Specific values == | == Specific values == | ||
| ^ $\zeta(-2m)=0$ ^ | ^ $\zeta(-2m)=0$ ^ | ||
| ^ $\zeta(2m)=(-1)^{m+1}\frac{(2\pi)^{2m}}{2(2m)!} B_{2m}$ ^ | ^ $\zeta(2m)=(-1)^{m+1}\frac{(2\pi)^{2m}}{2(2m)!} B_{2m}$ ^ | ||
| + | |||
| + | so that | ||
| + | |||
| + | $\zeta(1-2m) = \dfrac{2(2m-1)!}{(4\pi^2)^m}\cos(m\pi)\zeta(2m)$ | ||
| + | |||
| + | $\zeta(1-2m)=(-1)^{m+1}\frac{1}{2m}B_{2m}$ | ||
| + | |||
| E.g. | E.g. | ||
| ^ $\zeta(2)=\pi^2/6$ ^ | ^ $\zeta(2)=\pi^2/6$ ^ | ||
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| $\sum_{n=1}^\infty \frac{1}{n^s} = \frac{1} { \Gamma(s) } \int_0^\infty \frac{x^{s-1}} { {\mathrm e}^x-1} \, {\mathrm d}x $ | $\sum_{n=1}^\infty \frac{1}{n^s} = \frac{1} { \Gamma(s) } \int_0^\infty \frac{x^{s-1}} { {\mathrm e}^x-1} \, {\mathrm d}x $ | ||
| - | He takes the integral into the complex plane, where he $ \frac{1} { {\mathrm e}^x-1}$ diverges periodically in steps of $2\pi\,i$. | + | He takes the integral into the complex plane, where the $ \frac{1} { {\mathrm e}^x-1}$ diverges periodically in steps of $2\pi\,i$. |
| He discovers that the function obeys a reflection formula | He discovers that the function obeys a reflection formula | ||
