\RiemannZeta β’ @ β’ s = β n = 1 β 1 n s \RiemannZeta @ π superscript subscript π 1 1 superscript π π {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\sum_{n=1}^{\infty}% \frac{1}{n^{s}}}}} {\displaystyle \RiemannZeta@{s} = \sum_{n=1}^\infty \frac{1}{n^s} }
\RiemannZeta β’ @ β’ s = 1 1 - 2 - s β’ β n = 0 β 1 ( 2 β’ n + 1 ) s \RiemannZeta @ π 1 1 superscript 2 π superscript subscript π 0 1 superscript 2 π 1 π {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{1-2^{-s}}% \sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}}}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{1 - 2^{-s}} \sum_{n=0}^\infty \frac{1}{(2n+1)^s} }
\RiemannZeta β’ @ β’ s = 1 1 - 2 1 - s β’ β n = 1 β ( - 1 ) n - 1 n s \RiemannZeta @ π 1 1 superscript 2 1 π superscript subscript π 1 superscript 1 π 1 superscript π π {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{1-2^{1-s}}% \sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{1 - 2^{1-s}} \sum_{n=1}^\infty \frac{\opminus^{n-1}}{n^s} }
\RiemannZeta β’ @ β’ s = 1 s - 1 + β n = 0 β ( - 1 ) n n ! β’ Ξ³ n β’ ( s - 1 ) n \RiemannZeta @ π 1 π 1 superscript subscript π 0 superscript 1 π π Stieltjes-constants π superscript π 1 π {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{s-1}+\sum_% {n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma_{n}(s-1)^{n}}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{s-1} + \sum_{n=0}^\infty \frac{\opminus^n}{n!} \StieltjesConstants{n} (s-1)^n }
\RiemannZeta β² β’ @ β’ s = - β n = 2 β ( ln β‘ n ) β’ n - s superscript \RiemannZeta β² @ π superscript subscript π 2 π superscript π π {\displaystyle{\displaystyle{\displaystyle\RiemannZeta^{\prime}@{s}=-\sum_{n=2% }^{\infty}(\ln n)n^{-s}}}} {\displaystyle \RiemannZeta'@{s} = - \sum_{n=2}^\infty (\ln@@{n}) n^{-s} }
\RiemannZeta ( k ) β’ @ β’ s = ( - 1 ) k β’ β n = 2 β ( ln β‘ n ) k β’ n - s superscript \RiemannZeta π @ π superscript 1 π superscript subscript π 2 superscript π π superscript π π {\displaystyle{\displaystyle{\displaystyle\RiemannZeta^{(k)}@{s}=(-1)^{k}\sum_% {n=2}^{\infty}(\ln n)^{k}n^{-s}}}} {\displaystyle \RiemannZeta^{(k)}@{s} = \opminus^k \sum_{n=2}^\infty (\ln@@{n})^k n^{-s} }
\RiemannZeta β’ @ β’ s = β k = 1 N 1 k s + N 1 - s s - 1 - s β’ β« N β x - β x β x s + 1 β’ d x \RiemannZeta @ π superscript subscript π 1 π 1 superscript π π superscript π 1 π π 1 π superscript subscript π π₯ π₯ superscript π₯ π 1 π₯ {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\sum_{k=1}^{N}\frac% {1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right% \rfloor}{x^{s+1}}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = \sum_{k=1}^N \frac{1}{k^s} + \frac{N^{1-s}}{s-1} - s \int_N^\infty \frac{x-\floor{x}}{x^{s+1}} \diff{x} }
\RiemannZeta β’ @ β’ s = β k = 1 N 1 k s + N 1 - s s - 1 - 1 2 β’ N - s + β k = 1 n ( s + 2 β’ k - 2 2 β’ k - 1 ) β’ \BernoulliB β’ 2 β’ k 2 β’ k β’ N 1 - s - 2 β’ k - ( s + 2 β’ n 2 β’ n + 1 ) β’ β« N β \PeriodicBernoulliB β’ 2 β’ n + 1 β’ @ β’ x x s + 2 β’ n + 1 β’ d x \RiemannZeta @ π superscript subscript π 1 π 1 superscript π π superscript π 1 π π 1 1 2 superscript π π superscript subscript π 1 π binomial π 2 π 2 2 π 1 \BernoulliB 2 π 2 π superscript π 1 π 2 π binomial π 2 π 2 π 1 superscript subscript π \PeriodicBernoulliB 2 π 1 @ π₯ superscript π₯ π 2 π 1 π₯ {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\sum_{k=1}^{N}\frac% {1}{k^{s}}+\frac{N^{1-s}}{s-1}-\frac{1}{2}N^{-s}+\sum_{k=1}^{n}\genfrac{(}{)}{% 0.0pt}{}{s+2k-2}{2k-1}\frac{\BernoulliB{2k}}{2k}N^{1-s-2k}-\genfrac{(}{)}{0.0% pt}{}{s+2n}{2n+1}\int_{N}^{\infty}\frac{\PeriodicBernoulliB{2n+1}@{x}}{x^{s+2n% +1}}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = \sum_{k=1}^N \frac{1}{k^s} + \frac{N^{1-s}}{s-1} - \frac{1}{2}N^{-s} + \sum_{k=1}^n \binom{s+2k-2}{2k-1} \frac{\BernoulliB{2k}}{2k} N^{1-s-2k} - \binom{s+2n}{2n+1} \int_N^\infty \frac{\PeriodicBernoulliB{2n+1}@{x}}{x^{s+2n+1}} \diff{x} }
\RiemannZeta β’ @ β’ s = 1 s - 1 + 1 2 + β k = 1 n ( s + 2 β’ k - 2 2 β’ k - 1 ) β’ \BernoulliB β’ 2 β’ k 2 β’ k - ( s + 2 β’ n 2 β’ n + 1 ) β’ β« 1 β \PeriodicBernoulliB β’ 2 β’ n + 1 β’ @ β’ x x s + 2 β’ n + 1 β’ d x \RiemannZeta @ π 1 π 1 1 2 superscript subscript π 1 π binomial π 2 π 2 2 π 1 \BernoulliB 2 π 2 π binomial π 2 π 2 π 1 superscript subscript 1 \PeriodicBernoulliB 2 π 1 @ π₯ superscript π₯ π 2 π 1 π₯ {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{s-1}+\frac% {1}{2}+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{\BernoulliB{2k% }}{2k}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{1}^{\infty}\frac{% \PeriodicBernoulliB{2n+1}@{x}}{x^{s+2n+1}}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{s-1} + \frac{1}{2} + \sum_{k=1}^n \binom{s+2k-2}{2k-1} \frac{\BernoulliB{2k}}{2k} - \binom{s+2n}{2n+1} \int_1^\infty \frac{\PeriodicBernoulliB{2n+1}@{x}}{x^{s+2n+1}} \diff{x} }
\RiemannZeta β’ @ β’ s = β p ( 1 - p - s ) - 1 \RiemannZeta @ π subscript product π superscript 1 superscript π π 1 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\prod_{p}(1-p^{-s})% ^{-1}}}} {\displaystyle \RiemannZeta@{s} = \prod_p (1 - p^{-s})^{-1} }
\RiemannZeta β’ @ β’ s = ( 2 β’ Ο ) s β’ e - s - ( Ξ³ β’ s / 2 ) 2 β’ ( s - 1 ) β’ Ξ β‘ ( 1 2 β’ s + 1 ) β’ β Ο ( 1 - s Ο ) β’ e s / Ο \RiemannZeta @ π superscript 2 π π π 2 2 π 1 Euler-Gamma 1 2 π 1 subscript product π 1 π π π π {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{(2\pi)^{s}{% \mathrm{e}^{-s-(\gamma s/2)}}}{2(s-1)\Gamma\left(\tfrac{1}{2}s+1\right)}\prod_% {\rho}\left(1-\frac{s}{\rho}\right){\mathrm{e}^{s/\rho}}}}} {\displaystyle \RiemannZeta@{s} = \frac{(2 \cpi)^s \expe^{-s -(\EulerConstant s/2)}} {2(s-1) \EulerGamma@{\tfrac{1}{2} s + 1}} \prod_\rho \left( 1 - \frac{s}{\rho} \right) \expe^{s/\rho} }