Zeta and Related Functions - 25.8 Sums

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25.8.E1 ∑ k = 2 ∞ ( ζ ⁡ ( k ) - 1 ) = 1 superscript subscript 𝑘 2 Riemann-zeta 𝑘 1 1 {\displaystyle{\displaystyle\sum_{k=2}^{\infty}\left(\zeta\left(k\right)-1% \right)=1}}
\sum_{k=2}^{\infty}\left(\Riemannzeta@{k}-1\right) = 1		 

sum(Zeta(k)- 1, k = 2..infinity) = 1

Sum[Zeta[k]- 1, {k, 2, Infinity}, GenerateConditions->None] == 1
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
25.8.E2 ∑ k = 0 ∞ Γ ⁡ ( s + k ) ( k + 1 ) ! ⁢ ( ζ ⁡ ( s + k ) - 1 ) = Γ ⁡ ( s - 1 ) superscript subscript 𝑘 0 Euler-Gamma 𝑠 𝑘 𝑘 1 Riemann-zeta 𝑠 𝑘 1 Euler-Gamma 𝑠 1 {\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{\Gamma\left(s+k\right)}{(% k+1)!}\left(\zeta\left(s+k\right)-1\right)=\Gamma\left(s-1\right)}}
\sum_{k=0}^{\infty}\frac{\EulerGamma@{s+k}}{(k+1)!}\left(\Riemannzeta@{s+k}-1\right) = \EulerGamma@{s-1}		 ℜ ⁡ ( s + k ) > 0 , ℜ ⁡ ( s - 1 ) > 0 formulae-sequence 𝑠 𝑘 0 𝑠 1 0 {\displaystyle{\displaystyle\Re(s+k)>0,\Re(s-1)>0}} 
sum((GAMMA(s + k))/(factorial(k + 1))*(Zeta(s + k)- 1), k = 0..infinity) = GAMMA(s - 1)

Sum[Divide[Gamma[s + k],(k + 1)!]*(Zeta[s + k]- 1), {k, 0, Infinity}, GenerateConditions->None] == Gamma[s - 1]
Failure Aborted Successful [Tested: 1] Skipped - Because timed out
25.8.E3 ∑ k = 0 ∞ ( s ) k ⁢ ζ ⁡ ( s + k ) k ! ⁢ 2 s + k = ( 1 - 2 - s ) ⁢ ζ ⁡ ( s ) superscript subscript 𝑘 0 Pochhammer 𝑠 𝑘 Riemann-zeta 𝑠 𝑘 𝑘 superscript 2 𝑠 𝑘 1 superscript 2 𝑠 Riemann-zeta 𝑠 {\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{{\left(s\right)_{k}}\zeta% \left(s+k\right)}{k!2^{s+k}}=(1-2^{-s})\zeta\left(s\right)}}
\sum_{k=0}^{\infty}\frac{\Pochhammersym{s}{k}\Riemannzeta@{s+k}}{k!2^{s+k}} = (1-2^{-s})\Riemannzeta@{s}		 s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}} 
sum((pochhammer(s, k)*Zeta(s + k))/(factorial(k)*(2)^(s + k)), k = 0..infinity) = (1 - (2)^(- s))*Zeta(s)

Sum[Divide[Pochhammer[s, k]*Zeta[s + k],(k)!*(2)^(s + k)], {k, 0, Infinity}, GenerateConditions->None] == (1 - (2)^(- s))*Zeta[s]
Failure Successful
Failed [1 / 6]
Result: -.1666666667
Test Values: {s = -2}

Successful [Tested: 6]
25.8.E4 ∑ k = 1 ∞ ( - 1 ) k k ⁢ ( ζ ⁡ ( n ⁢ k ) - 1 ) = ln ⁡ ( ∏ j = 0 n - 1 Γ ⁡ ( 2 - e ( 2 ⁢ j + 1 ) ⁢ π ⁢ i / n ) ) superscript subscript 𝑘 1 superscript 1 𝑘 𝑘 Riemann-zeta 𝑛 𝑘 1 superscript subscript product 𝑗 0 𝑛 1 Euler-Gamma 2 superscript 𝑒 2 𝑗 1 𝜋 𝑖 𝑛 {\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-e^{(2j+1)\pi i/n}\right)% \right)}}
\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\Riemannzeta@{nk}-1) = \ln@{\prod_{j=0}^{n-1}\EulerGamma@{2-e^{(2j+1)\pi i/n}}}		 

sum(((- 1)^(k))/(k)*(Zeta(n*k)- 1), k = 1..infinity) = ln(product(GAMMA(2 - exp((2*j + 1)*Pi*I/n)), j = 0..n - 1))

Sum[Divide[(- 1)^(k),k]*(Zeta[n*k]- 1), {k, 1, Infinity}, GenerateConditions->None] == Log[Product[Gamma[2 - Exp[(2*j + 1)*Pi*I/n]], {j, 0, n - 1}, GenerateConditions->None]]
Failure Failure Successful [Tested: 1]
Failed [1 / 3]
Result: Plus[-0.6931471805599453, NSum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[k]]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[n, 1]}

25.8.E5 ∑ k = 2 ∞ ζ ⁡ ( k ) ⁢ z k = - γ ⁢ z - z ⁢ ψ ⁡ ( 1 - z ) superscript subscript 𝑘 2 Riemann-zeta 𝑘 superscript 𝑧 𝑘 𝑧 𝑧 digamma 1 𝑧 {\displaystyle{\displaystyle\sum_{k=2}^{\infty}\zeta\left(k\right)z^{k}=-% \gamma z-z\psi\left(1-z\right)}}
\sum_{k=2}^{\infty}\Riemannzeta@{k}z^{k} = -\EulerConstant z-z\digamma@{1-z}		 | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}} 
sum(Zeta(k)*(z)^(k), k = 2..infinity) = - gamma*z - z*Psi(1 - z)

Sum[Zeta[k]*(z)^(k), {k, 2, Infinity}, GenerateConditions->None] == - EulerGamma*z - z*PolyGamma[1 - z]
Failure Successful Successful [Tested: 1] Successful [Tested: 1]
25.8.E6 ∑ k = 0 ∞ ζ ⁡ ( 2 ⁢ k ) ⁢ z 2 ⁢ k = - 1 2 ⁢ π ⁢ z ⁢ cot ⁡ ( π ⁢ z ) superscript subscript 𝑘 0 Riemann-zeta 2 𝑘 superscript 𝑧 2 𝑘 1 2 𝜋 𝑧 𝜋 𝑧 {\displaystyle{\displaystyle\sum_{k=0}^{\infty}\zeta\left(2k\right)z^{2k}=-% \tfrac{1}{2}\pi z\cot\left(\pi z\right)}}
\sum_{k=0}^{\infty}\Riemannzeta@{2k}z^{2k} = -\tfrac{1}{2}\pi z\cot@{\pi z}		 | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}} 
sum(Zeta(2*k)*(z)^(2*k), k = 0..infinity) = -(1)/(2)*Pi*z*cot(Pi*z)

Sum[Zeta[2*k]*(z)^(2*k), {k, 0, Infinity}, GenerateConditions->None] == -Divide[1,2]*Pi*z*Cot[Pi*z]
Failure Failure Error
Failed [1 / 1]
Result: Plus[4.8091767343044744*^-17, NSum[Times[Power[0.5, Times[2, k]], Zeta[Times[2, k]]]
Test Values: {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[z, 0.5]}

25.8.E7 ∑ k = 2 ∞ ζ ⁡ ( k ) k ⁢ z k = - γ ⁢ z + ln ⁡ Γ ⁡ ( 1 - z ) superscript subscript 𝑘 2 Riemann-zeta 𝑘 𝑘 superscript 𝑧 𝑘 𝑧 Euler-Gamma 1 𝑧 {\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{\zeta\left(k\right)}{k}z^% {k}=-\gamma z+\ln\Gamma\left(1-z\right)}}
\sum_{k=2}^{\infty}\frac{\Riemannzeta@{k}}{k}z^{k} = -\EulerConstant z+\ln@@{\EulerGamma@{1-z}}		 | z | < 1 , ℜ ⁡ ( 1 - z ) > 0 formulae-sequence 𝑧 1 1 𝑧 0 {\displaystyle{\displaystyle|z|<1,\Re(1-z)>0}} 
sum((Zeta(k))/(k)*(z)^(k), k = 2..infinity) = - gamma*z + ln(GAMMA(1 - z))

Sum[Divide[Zeta[k],k]*(z)^(k), {k, 2, Infinity}, GenerateConditions->None] == - EulerGamma*z + Log[Gamma[1 - z]]
Failure Successful Successful [Tested: 1] Successful [Tested: 1]
25.8.E8 ∑ k = 1 ∞ ζ ⁡ ( 2 ⁢ k ) k ⁢ z 2 ⁢ k = ln ⁡ ( π ⁢ z sin ⁡ ( π ⁢ z ) ) superscript subscript 𝑘 1 Riemann-zeta 2 𝑘 𝑘 superscript 𝑧 2 𝑘 𝜋 𝑧 𝜋 𝑧 {\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}z% ^{2k}=\ln\left(\frac{\pi z}{\sin\left(\pi z\right)}\right)}}
\sum_{k=1}^{\infty}\frac{\Riemannzeta@{2k}}{k}z^{2k} = \ln@{\frac{\pi z}{\sin@{\pi z}}}		 | z | < 1 𝑧 1 {\displaystyle{\displaystyle|z|<1}} 
sum((Zeta(2*k))/(k)*(z)^(2*k), k = 1..infinity) = ln((Pi*z)/(sin(Pi*z)))

Sum[Divide[Zeta[2*k],k]*(z)^(2*k), {k, 1, Infinity}, GenerateConditions->None] == Log[Divide[Pi*z,Sin[Pi*z]]]
Failure Successful Successful [Tested: 1] Successful [Tested: 1]
25.8.E9 ∑ k = 1 ∞ ζ ⁡ ( 2 ⁢ k ) ( 2 ⁢ k + 1 ) ⁢ 2 2 ⁢ k = 1 2 - 1 2 ⁢ ln ⁡ 2 superscript subscript 𝑘 1 Riemann-zeta 2 𝑘 2 𝑘 1 superscript 2 2 𝑘 1 2 1 2 2 {\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k% +1)2^{2k}}=\frac{1}{2}-\frac{1}{2}\ln 2}}
\sum_{k=1}^{\infty}\frac{\Riemannzeta@{2k}}{(2k+1)2^{2k}} = \frac{1}{2}-\frac{1}{2}\ln@@{2}		 

sum((Zeta(2*k))/((2*k + 1)*(2)^(2*k)), k = 1..infinity) = (1)/(2)-(1)/(2)*ln(2)

Sum[Divide[Zeta[2*k],(2*k + 1)*(2)^(2*k)], {k, 1, Infinity}, GenerateConditions->None] == Divide[1,2]-Divide[1,2]*Log[2]
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
25.8.E10 ∑ k = 1 ∞ ζ ⁡ ( 2 ⁢ k ) ( 2 ⁢ k + 1 ) ⁢ ( 2 ⁢ k + 2 ) ⁢ 2 2 ⁢ k = 1 4 - 7 4 ⁢ π 2 ⁢ ζ ⁡ ( 3 ) superscript subscript 𝑘 1 Riemann-zeta 2 𝑘 2 𝑘 1 2 𝑘 2 superscript 2 2 𝑘 1 4 7 4 superscript 𝜋 2 Riemann-zeta 3 {\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k% +1)(2k+2)2^{2k}}=\frac{1}{4}-\frac{7}{4\pi^{2}}\zeta\left(3\right)}}
\sum_{k=1}^{\infty}\frac{\Riemannzeta@{2k}}{(2k+1)(2k+2)2^{2k}} = \frac{1}{4}-\frac{7}{4\pi^{2}}\Riemannzeta@{3}		 

sum((Zeta(2*k))/((2*k + 1)*(2*k + 2)*(2)^(2*k)), k = 1..infinity) = (1)/(4)-(7)/(4*(Pi)^(2))*Zeta(3)

Sum[Divide[Zeta[2*k],(2*k + 1)*(2*k + 2)*(2)^(2*k)], {k, 1, Infinity}, GenerateConditions->None] == Divide[1,4]-Divide[7,4*(Pi)^(2)]*Zeta[3]
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
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