Sums

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Sums

βˆ‘ k = 2 ∞ ( \RiemannZeta ⁒ @ ⁒ k - 1 ) = 1 superscript subscript π‘˜ 2 \RiemannZeta @ π‘˜ 1 1 {\displaystyle{\displaystyle{\displaystyle\sum_{k=2}^{\infty}\left(% \RiemannZeta@{k}-1\right)=1}}} {\displaystyle \sum_{k \hiderel{=} 2}^\infty \left( \RiemannZeta@{k} - 1 \right) = 1 }
βˆ‘ k = 0 ∞ Ξ“ ⁑ ( s + k ) ( k + 1 ) ! ⁒ ( \RiemannZeta ⁒ @ ⁒ s + k - 1 ) = Ξ“ ⁑ ( s - 1 ) superscript subscript π‘˜ 0 Euler-Gamma 𝑠 π‘˜ π‘˜ 1 \RiemannZeta @ 𝑠 π‘˜ 1 Euler-Gamma 𝑠 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} }

Constraint(s): s β‰  1 , 0 , - 1 , - 2 , … 𝑠 1 0 1 2 … {\displaystyle{\displaystyle{\displaystyle s\neq 1,0,-1,-2,\dots}}}


βˆ‘ k = 0 ∞ Ξ“ ⁑ ( s + k ) ⁒ \RiemannZeta ⁒ @ ⁒ s + k k ! ⁒ Ξ“ ⁑ ( s ) ⁒ 2 s + k = ( 1 - 2 - s ) ⁒ \RiemannZeta ⁒ @ ⁒ s superscript subscript π‘˜ 0 Euler-Gamma 𝑠 π‘˜ \RiemannZeta @ 𝑠 π‘˜ π‘˜ Euler-Gamma 𝑠 superscript 2 𝑠 π‘˜ 1 superscript 2 𝑠 \RiemannZeta @ 𝑠 {\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} }

Constraint(s): s β‰  1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}}


βˆ‘ k = 1 ∞ ( - 1 ) k k ⁒ ( \RiemannZeta ⁒ @ ⁒ n ⁒ k - 1 ) = ln ⁑ ( ∏ j = 0 n - 1 Ξ“ ⁑ ( 2 - e ( 2 ⁒ j + 1 ) ⁒ Ο€ ⁒ i / n ) ) superscript subscript π‘˜ 1 superscript 1 π‘˜ π‘˜ \RiemannZeta @ 𝑛 π‘˜ 1 superscript subscript product 𝑗 0 𝑛 1 Euler-Gamma 2 2 𝑗 1 imaginary-unit 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k% }(\RiemannZeta@{nk}-1)=\ln\left(\prod_{j=0}^{n-1}\Gamma\left(2-{\mathrm{e}^{(2% j+1)\pi\mathrm{i}/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}}} }

Constraint(s): n = 2 , 3 , 4 , … 𝑛 2 3 4 … {\displaystyle{\displaystyle{\displaystyle n=2,3,4,\dots}}}


βˆ‘ k = 2 ∞ \RiemannZeta ⁒ @ ⁒ k ⁒ z k = - Ξ³ ⁒ z - z ⁒ ψ ⁑ ( 1 - z ) superscript subscript π‘˜ 2 \RiemannZeta @ π‘˜ superscript 𝑧 π‘˜ 𝑧 𝑧 digamma 1 𝑧 {\displaystyle{\displaystyle{\displaystyle\sum_{k=2}^{\infty}\RiemannZeta@{k}z% ^{k}=-\gamma z-z\psi\left(1-z\right)}}} {\displaystyle \sum_{k \hiderel{=} 2}^\infty \RiemannZeta@{k} z^k = - \EulerConstant z - z \digamma@{1-z} }

Constraint(s): | z | < 1 𝑧 1 {\displaystyle{\displaystyle{\displaystyle|z|<1}}}


βˆ‘ k = 0 ∞ \RiemannZeta ⁒ @ ⁒ 2 ⁒ k ⁒ z 2 ⁒ k = - 1 2 ⁒ Ο€ ⁒ z ⁒ cot ⁑ ( Ο€ ⁒ z ) superscript subscript π‘˜ 0 \RiemannZeta @ 2 π‘˜ superscript 𝑧 2 π‘˜ 1 2 𝑧 𝑧 {\displaystyle{\displaystyle{\displaystyle\sum_{k=0}^{\infty}\RiemannZeta@{2k}% z^{2k}=-\tfrac{1}{2}\pi z\cot\left(\pi z\right)}}} {\displaystyle \sum_{k \hiderel{=} 0}^\infty \RiemannZeta@{2k} z^{2k} = - \tfrac{1}{2} \cpi z \cot@{\cpi z} }

Constraint(s): | z | < 1 𝑧 1 {\displaystyle{\displaystyle{\displaystyle|z|<1}}}


βˆ‘ k = 2 ∞ \RiemannZeta ⁒ @ ⁒ k k ⁒ z k = - Ξ³ ⁒ z + ln ⁑ Ξ“ ⁑ ( 1 - z ) superscript subscript π‘˜ 2 \RiemannZeta @ π‘˜ π‘˜ superscript 𝑧 π‘˜ 𝑧 Euler-Gamma 1 𝑧 {\displaystyle{\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{% \RiemannZeta@{k}}{k}z^{k}=-\gamma 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}} }

Constraint(s): | z | < 1 𝑧 1 {\displaystyle{\displaystyle{\displaystyle|z|<1}}}


βˆ‘ k = 1 ∞ \RiemannZeta ⁒ @ ⁒ 2 ⁒ k k ⁒ z 2 ⁒ k = ln ⁑ ( Ο€ ⁒ z sin ⁑ ( Ο€ ⁒ z ) ) superscript subscript π‘˜ 1 \RiemannZeta @ 2 π‘˜ π‘˜ superscript 𝑧 2 π‘˜ 𝑧 𝑧 {\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}}} }

Constraint(s): | z | < 1 𝑧 1 {\displaystyle{\displaystyle{\displaystyle|z|<1}}}


βˆ‘ k = 1 ∞ \RiemannZeta ⁒ @ ⁒ 2 ⁒ k ( 2 ⁒ k + 1 ) ⁒ 2 2 ⁒ k = 1 2 - 1 2 ⁒ ln ⁑ 2 superscript subscript π‘˜ 1 \RiemannZeta @ 2 π‘˜ 2 π‘˜ 1 superscript 2 2 π‘˜ 1 2 1 2 2 {\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{% \RiemannZeta@{2k}}{(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 }
βˆ‘ k = 1 ∞ \RiemannZeta ⁒ @ ⁒ 2 ⁒ k ( 2 ⁒ k + 1 ) ⁒ ( 2 ⁒ k + 2 ) ⁒ 2 2 ⁒ k = 1 4 - 7 4 ⁒ Ο€ 2 ⁒ \RiemannZeta ⁒ @ ⁒ 3 superscript subscript π‘˜ 1 \RiemannZeta @ 2 π‘˜ 2 π‘˜ 1 2 π‘˜ 2 superscript 2 2 π‘˜ 1 4 7 4 2 \RiemannZeta @ 3 {\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{% \RiemannZeta@{2k}}{(2k+1)(2k+2)2^{2k}}=\frac{1}{4}-\frac{7}{4{\pi^{2}}}% \RiemannZeta@{3}}}} {\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} }

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