β k = 2 β ( ΞΆ β‘ ( k ) - 1 ) = 1 superscript subscript π 2 Riemann-zeta π 1 1 {\displaystyle{\displaystyle{\displaystyle\sum_{k=2}^{\infty}\left(\zeta\left(% k\right)-1\right)=1}}} {\displaystyle \sum_{k \hiderel{=} 2}^\infty \left( \RiemannZeta@{k} - 1 \right) = 1 } β k = 0 β Ξ β‘ ( s + k ) ( k + 1 ) ! β’ ( \RiemannZeta β’ @ β’ s + k - 1 ) = Ξ β‘ ( s - 1 ) superscript subscript π 0 Euler-Gamma π π π 1 \RiemannZeta @ π π 1 Euler-Gamma π 1 {\displaystyle{\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{\Gamma\left% (s+k\right)}{(k+1)!}\left(\RiemannZeta@{s+k}-1\right)=\Gamma\left(s-1\right)}}} {\displaystyle \sum_{k \hiderel{=} 0}^\infty \frac{\EulerGamma@{s+k}}{(k+1)!} \left( \RiemannZeta@{s+k} - 1 \right) = \EulerGamma@{s-1} }
β k = 0 β Ξ β‘ ( s + k ) β’ \RiemannZeta β’ @ β’ s + k k ! β’ Ξ β‘ ( s ) β’ 2 s + k = ( 1 - 2 - s ) β’ \RiemannZeta β’ @ β’ s superscript subscript π 0 Euler-Gamma π π \RiemannZeta @ π π π Euler-Gamma π superscript 2 π π 1 superscript 2 π \RiemannZeta @ π {\displaystyle{\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{\Gamma\left% (s+k\right)\RiemannZeta@{s+k}}{k!\Gamma\left(s\right)2^{s+k}}=(1-2^{-s})% \RiemannZeta@{s}}}} {\displaystyle \sum_{k \hiderel{=} 0}^\infty \frac{\EulerGamma@{s+k} \RiemannZeta@{s+k}}{k! \EulerGamma@{s} 2^{s+k}} = (1 - 2^{-s}) \RiemannZeta@{s} }
β k = 1 β ( - 1 ) k k β’ ( ΞΆ β‘ ( n β’ k ) - 1 ) = ln β‘ ( β j = 0 n - 1 Ξ β‘ ( 2 - \expe ( 2 β’ j + 1 ) β’ \cpi β’ \iunit / n ) ) superscript subscript π 1 superscript 1 π π Riemann-zeta π π 1 superscript subscript product π 0 π 1 Euler-Gamma 2 superscript \expe 2 π 1 \cpi \iunit π {\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k% }(\zeta\left(nk\right)-1)=\ln\!\left(\prod_{j=0}^{n-1}\Gamma\left(2-\expe^{(2j% +1)\cpi\iunit/n}\right)\right)}}} {\displaystyle \sum_{k \hiderel{=} 1}^\infty \frac{\opminus^k}{k} (\RiemannZeta@{nk} - 1) = \ln@{\prod_{j=0}^{n-1} \EulerGamma@{2 - \expe^{(2j+1) \cpi \iunit/n}}} }
β k = 2 β ΞΆ β‘ ( k ) β’ z k = - \EulerConstant β’ z - z β’ Ο β‘ ( 1 - z ) superscript subscript π 2 Riemann-zeta π superscript π§ π \EulerConstant π§ π§ digamma 1 π§ {\displaystyle{\displaystyle{\displaystyle\sum_{k=2}^{\infty}\zeta\left(k% \right)z^{k}=-\EulerConstant z-z\psi\left(1-z\right)}}} {\displaystyle \sum_{k \hiderel{=} 2}^\infty \RiemannZeta@{k} z^k = - \EulerConstant z - z \digamma@{1-z} }
β k = 0 β ΞΆ β‘ ( 2 β’ k ) β’ z 2 β’ k = - 1 2 β’ \cpi β’ z β’ cot β‘ ( \cpi β’ z ) superscript subscript π 0 Riemann-zeta 2 π superscript π§ 2 π 1 2 \cpi π§ \cpi π§ {\displaystyle{\displaystyle{\displaystyle\sum_{k=0}^{\infty}\zeta\left(2k% \right)z^{2k}=-\tfrac{1}{2}\cpi z\cot\!\left(\cpi z\right)}}} {\displaystyle \sum_{k \hiderel{=} 0}^\infty \RiemannZeta@{2k} z^{2k} = - \tfrac{1}{2} \cpi z \cot@{\cpi z} }
β k = 2 β ΞΆ β‘ ( k ) k β’ z k = - \EulerConstant β’ z + ln β‘ Ξ β‘ ( 1 - z ) superscript subscript π 2 Riemann-zeta π π superscript π§ π \EulerConstant π§ Euler-Gamma 1 π§ {\displaystyle{\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{\zeta\left(% k\right)}{k}z^{k}=-\EulerConstant z+\ln\Gamma\left(1-z\right)}}} {\displaystyle \sum_{k \hiderel{=} 2}^\infty \frac{\RiemannZeta@{k}}{k} z^k = -\EulerConstant z + \ln@@{\EulerGamma@{1-z}} }
β k = 1 β \RiemannZeta β’ @ β’ 2 β’ k k β’ z 2 β’ k = ln β‘ ( Ο β’ z sin β‘ ( Ο β’ z ) ) superscript subscript π 1 \RiemannZeta @ 2 π π superscript π§ 2 π π§ π§ {\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{% \RiemannZeta@{2k}}{k}z^{2k}=\ln\left(\frac{\pi z}{\sin\left(\pi z\right)}% \right)}}} {\displaystyle \sum_{k \hiderel{=} 1}^\infty \frac{\RiemannZeta@{2k}}{k} z^{2k} = \ln@{\frac{\cpi z}{\sin@{\cpi z}}} }
β k = 1 β ΞΆ β‘ ( 2 β’ k ) ( 2 β’ k + 1 ) β’ 2 2 β’ k = 1 2 - 1 2 β’ ln β‘ 2 superscript subscript π 1 Riemann-zeta 2 π 2 π 1 superscript 2 2 π 1 2 1 2 2 {\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(% 2k\right)}{(2k+1)2^{2k}}=\frac{1}{2}-\frac{1}{2}\ln 2}}} {\displaystyle \sum_{k \hiderel{=} 1}^\infty \frac{\RiemannZeta@{2k}}{(2k+1)2^{2k}} = \frac{1}{2} - \frac{1}{2} \ln 2 } β k = 1 β ΞΆ β‘ ( 2 β’ k ) ( 2 β’ k + 1 ) β’ ( 2 β’ k + 2 ) β’ 2 2 β’ k = 1 4 - 7 4 β’ \cpi 2 β’ ΞΆ β‘ ( 3 ) superscript subscript π 1 Riemann-zeta 2 π 2 π 1 2 π 2 superscript 2 2 π 1 4 7 4 superscript \cpi 2 Riemann-zeta 3 {\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(% 2k\right)}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}-\frac{7}{4\cpi^{2}}\zeta\left(3% \right)}}} {\displaystyle \sum_{k \hiderel{=} 1}^\infty \frac{\RiemannZeta@{2k}}{(2k+1) (2k+2) 2^{2k}} = \frac{1}{4} - \frac{7}{4 \cpi^2} \RiemannZeta@{3} }