View source for 25.6

{{DISPLAYTITLE:Zeta and Related Functions - 25.6 Integer Arguments}}

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! scope="col" style="position: sticky; top: 0;" | DLMF 
! scope="col" style="position: sticky; top: 0;" | Formula 
! scope="col" style="position: sticky; top: 0;" | Constraints 
! scope="col" style="position: sticky; top: 0;" | Maple
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! scope="col" style="position: sticky; top: 0;" | Symbolic<br>Maple
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! scope="col" style="position: sticky; top: 0;" | Numeric<br>Maple
! scope="col" style="position: sticky; top: 0;" | Numeric<br>Mathematica
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| [https://dlmf.nist.gov/25.6#Ex1 25.6#Ex1] || <math qid="Q7635">\Riemannzeta@{0} = -\frac{1}{2}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{0} = -\frac{1}{2}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(0) = -(1)/(2)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[0] == -Divide[1,2]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6#Ex2 25.6#Ex2] || <math qid="Q7636">\Riemannzeta@{2} = \frac{\pi^{2}}{6}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{2} = \frac{\pi^{2}}{6}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(2) = ((Pi)^(2))/(6)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[2] == Divide[(Pi)^(2),6]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6#Ex3 25.6#Ex3] || <math qid="Q7637">\Riemannzeta@{4} = \frac{\pi^{4}}{90}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{4} = \frac{\pi^{4}}{90}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(4) = ((Pi)^(4))/(90)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[4] == Divide[(Pi)^(4),90]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6#Ex4 25.6#Ex4] || <math qid="Q7638">\Riemannzeta@{6} = \frac{\pi^{6}}{945}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{6} = \frac{\pi^{6}}{945}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(6) = ((Pi)^(6))/(945)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[6] == Divide[(Pi)^(6),945]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6.E2 25.6.E2] || <math qid="Q7639">\Riemannzeta@{2n} = \frac{(2\pi)^{2n}}{2(2n)!}\left|\BernoullinumberB{2n}\right|</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{2n} = \frac{(2\pi)^{2n}}{2(2n)!}\left|\BernoullinumberB{2n}\right|</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(2*n) = ((2*Pi)^(2*n))/(2*factorial(2*n))*abs(bernoulli(2*n))</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[2*n] == Divide[(2*Pi)^(2*n),2*(2*n)!]*Abs[BernoulliB[2*n]]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6.E3 25.6.E3] || <math qid="Q7640">\Riemannzeta@{-n} = -\frac{\BernoullinumberB{n+1}}{n+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{-n} = -\frac{\BernoullinumberB{n+1}}{n+1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(- n) = -(bernoulli(n + 1))/(n + 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[- n] == -Divide[BernoulliB[n + 1],n + 1]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6.E4 25.6.E4] || <math qid="Q7641">\Riemannzeta@{-2n} = 0</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{-2n} = 0</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(- 2*n) = 0</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[- 2*n] == 0</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6.E6 25.6.E6] || <math qid="Q7643">\Riemannzeta@{2k+1} = \frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}\BernoullipolyB{2k+1}@{t}\cot@{\pi t}\diff{t}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{2k+1} = \frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}\BernoullipolyB{2k+1}@{t}\cot@{\pi t}\diff{t}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(2*k + 1) = ((- 1)^(k + 1)*(2*Pi)^(2*k + 1))/(2*factorial(2*k + 1))*int(bernoulli(2*k + 1, t)*cot(Pi*t), t = 0..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[2*k + 1] == Divide[(- 1)^(k + 1)*(2*Pi)^(2*k + 1),2*(2*k + 1)!]*Integrate[BernoulliB[2*k + 1, t]*Cot[Pi*t], {t, 0, 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6.E7 25.6.E7] || <math qid="Q7644">\Riemannzeta@{2} = \int_{0}^{1}\int_{0}^{1}\frac{1}{1-xy}\diff{x}\diff{y}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{2} = \int_{0}^{1}\int_{0}^{1}\frac{1}{1-xy}\diff{x}\diff{y}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(2) = int(int((1)/(1 - x*y), x = 0..1), y = 0..1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[2] == Integrate[Integrate[Divide[1,1 - x*y], {x, 0, 1}, GenerateConditions->None], {y, 0, 1}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6.E8 25.6.E8] || <math qid="Q7645">\Riemannzeta@{2} = 3\sum_{k=1}^{\infty}\frac{1}{k^{2}\binom{2k}{k}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{2} = 3\sum_{k=1}^{\infty}\frac{1}{k^{2}\binom{2k}{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(2) = 3*sum((1)/((k)^(2)*binomial(2*k,k)), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[2] == 3*Sum[Divide[1,(k)^(2)*Binomial[2*k,k]], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6.E9 25.6.E9] || <math qid="Q7646">\Riemannzeta@{3} = \frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^{3}\binom{2k}{k}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{3} = \frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^{3}\binom{2k}{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(3) = (5)/(2)*sum(((- 1)^(k - 1))/((k)^(3)*binomial(2*k,k)), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[3] == Divide[5,2]*Sum[Divide[(- 1)^(k - 1),(k)^(3)*Binomial[2*k,k]], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6.E10 25.6.E10] || <math qid="Q7647">\Riemannzeta@{4} = \frac{36}{17}\sum_{k=1}^{\infty}\frac{1}{k^{4}\binom{2k}{k}}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta@{4} = \frac{36}{17}\sum_{k=1}^{\infty}\frac{1}{k^{4}\binom{2k}{k}}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>Zeta(4) = (36)/(17)*sum((1)/((k)^(4)*binomial(2*k,k)), k = 1..infinity)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Zeta[4] == Divide[36,17]*Sum[Divide[1,(k)^(4)*Binomial[2*k,k]], {k, 1, Infinity}, GenerateConditions->None]</syntaxhighlight> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6.E11 25.6.E11] || <math qid="Q7648">\Riemannzeta'@{0} = -\tfrac{1}{2}\ln@{2\pi}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta'@{0} = -\tfrac{1}{2}\ln@{2\pi}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>subs( temp=0, diff( Zeta(temp), temp$(1) ) ) = -(1)/(2)*ln(2*Pi)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Zeta[temp], {temp, 1}]/.temp-> 0) == -Divide[1,2]*Log[2*Pi]</syntaxhighlight> || Successful || Successful || - || Successful [Tested: 1]
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| [https://dlmf.nist.gov/25.6.E12 25.6.E12] || <math qid="Q7649">\Riemannzeta''@{0} = -\tfrac{1}{2}(\ln@{2\pi})^{2}+\tfrac{1}{2}\EulerConstant^{2}-\tfrac{1}{24}\pi^{2}+\StieltjesConstants{1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta''@{0} = -\tfrac{1}{2}(\ln@{2\pi})^{2}+\tfrac{1}{2}\EulerConstant^{2}-\tfrac{1}{24}\pi^{2}+\StieltjesConstants{1}</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>subs( temp=0, diff( Zeta(temp), temp$(2) ) ) = -(1)/(2)*(ln(2*Pi))^(2)+(1)/(2)*(gamma)^(2)-(1)/(24)*(Pi)^(2)+ gamma(1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Error</syntaxhighlight> || Successful || Missing Macro Error || - || -
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| [https://dlmf.nist.gov/25.6.E13 25.6.E13] || <math qid="Q7650">(-1)^{k}\Riemannzeta^{(k)}@{-2n} = \frac{2(-1)^{n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{k}{m}\binom{m}{r}\imagpart@{c^{k-m}}\*\EulerGamma^{(r)}@{2n+1}\Riemannzeta^{(m-r)}@{2n+1}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{k}\Riemannzeta^{(k)}@{-2n} = \frac{2(-1)^{n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{k}{m}\binom{m}{r}\imagpart@{c^{k-m}}\*\EulerGamma^{(r)}@{2n+1}\Riemannzeta^{(m-r)}@{2n+1}</syntaxhighlight> || <math>\realpart@@{(2n+1)} > 0</math> || <syntaxhighlight lang=mathematica>(- 1)^(k)* subs( temp=- 2*n, diff( Zeta(temp), temp$(k) ) ) = (2*(- 1)^(n))/((2*Pi)^(2*n + 1))*sum(sum(binomial(k,m)*binomial(m,r)*Im((c)^(k - m))* subs( temp=2*n + 1, diff( GAMMA(temp), temp$(r) ) )*subs( temp=2*n + 1, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> - 2*n) == Divide[2*(- 1)^(n),(2*Pi)^(2*n + 1)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*Im[(c)^(k - m)]* (D[Gamma[temp], {temp, r}]/.temp-> 2*n + 1)*(D[Zeta[temp], {temp, m - r}]/.temp-> 2*n + 1), {r, 0, m}, GenerateConditions->None], {m, 0, k}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [48 / 54]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: 0.030448457058393275
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: -0.007983811450268627
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[n, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/25.6.E14 25.6.E14] || <math qid="Q7651">(-1)^{k}\Riemannzeta^{(k)}@{1-2n} = \frac{2(-1)^{n}}{(2\pi)^{2n}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{k}{m}\binom{m}{r}\realpart@{c^{k-m}}\*\EulerGamma^{(r)}@{2n}\Riemannzeta^{(m-r)}@{2n}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>(-1)^{k}\Riemannzeta^{(k)}@{1-2n} = \frac{2(-1)^{n}}{(2\pi)^{2n}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{k}{m}\binom{m}{r}\realpart@{c^{k-m}}\*\EulerGamma^{(r)}@{2n}\Riemannzeta^{(m-r)}@{2n}</syntaxhighlight> || <math>\realpart@@{(2n)} > 0</math> || <syntaxhighlight lang=mathematica>(- 1)^(k)* subs( temp=1 - 2*n, diff( Zeta(temp), temp$(k) ) ) = (2*(- 1)^(n))/((2*Pi)^(2*n))*sum(sum(binomial(k,m)*binomial(m,r)*Re((c)^(k - m))* subs( temp=2*n, diff( GAMMA(temp), temp$(r) ) )*subs( temp=2*n, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> 1 - 2*n) == Divide[2*(- 1)^(n),(2*Pi)^(2*n)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*Re[(c)^(k - m)]* (D[Gamma[temp], {temp, r}]/.temp-> 2*n)*(D[Zeta[temp], {temp, m - r}]/.temp-> 2*n), {r, 0, m}, GenerateConditions->None], {m, 0, k}, GenerateConditions->None]</syntaxhighlight> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><syntaxhighlight lang=mathematica>Result: Plus[0.1654211437004509, Times[0.05066059182116889, Plus[Times[-1.0772156649015328, D[1.6449340668482262
Test Values: {2.0, 0.0}]], Times[1.0, D[1.6449340668482262, {2.0, 1.0}]]]]], {Rule[c, -1.5], Rule[k, 1], Rule[n, 1]}</syntaxhighlight><br><syntaxhighlight lang=mathematica>Result: Plus[-0.005378576357774297, Times[-0.001283247781835542, Plus[Times[-1.4632939894091983, D[1.0823232337111381
Test Values: {4.0, 0.0}]], Times[6.0, D[1.0823232337111381, {4.0, 1.0}]]]]], {Rule[c, -1.5], Rule[k, 1], Rule[n, 2]}</syntaxhighlight><br>... skip entries to safe data</div></div>
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| [https://dlmf.nist.gov/25.6.E15 25.6.E15] || <math qid="Q7652">\Riemannzeta'@{2n} = \frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\Riemannzeta'@{1-2n}-(\digamma@{2n}-\ln@{2\pi})\BernoullinumberB{2n}\right)</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\Riemannzeta'@{2n} = \frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\Riemannzeta'@{1-2n}-(\digamma@{2n}-\ln@{2\pi})\BernoullinumberB{2n}\right)</syntaxhighlight> || <math></math> || <syntaxhighlight lang=mathematica>subs( temp=2*n, diff( Zeta(temp), temp$(1) ) ) = ((- 1)^(n + 1)*(2*Pi)^(2*n))/(2*factorial(2*n))*(2*n*subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) )-(Psi(2*n)- ln(2*Pi))*bernoulli(2*n))</syntaxhighlight> || <syntaxhighlight lang=mathematica>(D[Zeta[temp], {temp, 1}]/.temp-> 2*n) == Divide[(- 1)^(n + 1)*(2*Pi)^(2*n),2*(2*n)!]*(2*n*(D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n)-(PolyGamma[2*n]- Log[2*Pi])*BernoulliB[2*n])</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/25.6.E16 25.6.E16] || <math qid="Q7653">\left(n+\tfrac{1}{2}\right)\Riemannzeta@{2n} = \sum_{k=1}^{n-1}\Riemannzeta@{2k}\Riemannzeta@{2n-2k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(n+\tfrac{1}{2}\right)\Riemannzeta@{2n} = \sum_{k=1}^{n-1}\Riemannzeta@{2k}\Riemannzeta@{2n-2k}</syntaxhighlight> || <math>n \geq 2</math> || <syntaxhighlight lang=mathematica>(n +(1)/(2))*Zeta(2*n) = sum(Zeta(2*k)*Zeta(2*n - 2*k), k = 1..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(n +Divide[1,2])*Zeta[2*n] == Sum[Zeta[2*k]*Zeta[2*n - 2*k], {k, 1, n - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
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| [https://dlmf.nist.gov/25.6.E17 25.6.E17] || <math qid="Q7654">\left(n+\tfrac{3}{4}\right)\Riemannzeta@{4n+2} = \sum_{k=1}^{n}\Riemannzeta@{2k}\Riemannzeta@{4n+2-2k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(n+\tfrac{3}{4}\right)\Riemannzeta@{4n+2} = \sum_{k=1}^{n}\Riemannzeta@{2k}\Riemannzeta@{4n+2-2k}</syntaxhighlight> || <math>n \geq 1</math> || <syntaxhighlight lang=mathematica>(n +(3)/(4))*Zeta(4*n + 2) = sum(Zeta(2*k)*Zeta(4*n + 2 - 2*k), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(n +Divide[3,4])*Zeta[4*n + 2] == Sum[Zeta[2*k]*Zeta[4*n + 2 - 2*k], {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| [https://dlmf.nist.gov/25.6.E18 25.6.E18] || <math qid="Q7655">\left(n+\tfrac{1}{4}\right)\Riemannzeta@{4n}+\tfrac{1}{2}(\Riemannzeta@{2n})^{2} = \sum_{k=1}^{n}\Riemannzeta@{2k}\Riemannzeta@{4n-2k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(n+\tfrac{1}{4}\right)\Riemannzeta@{4n}+\tfrac{1}{2}(\Riemannzeta@{2n})^{2} = \sum_{k=1}^{n}\Riemannzeta@{2k}\Riemannzeta@{4n-2k}</syntaxhighlight> || <math>n \geq 1</math> || <syntaxhighlight lang=mathematica>(n +(1)/(4))*Zeta(4*n)+(1)/(2)*(Zeta(2*n))^(2) = sum(Zeta(2*k)*Zeta(4*n - 2*k), k = 1..n)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(n +Divide[1,4])*Zeta[4*n]+Divide[1,2]*(Zeta[2*n])^(2) == Sum[Zeta[2*k]*Zeta[4*n - 2*k], {k, 1, n}, GenerateConditions->None]</syntaxhighlight> || Skipped - Unable to analyze test case: Null || Skipped - Unable to analyze test case: Null || - || -
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| [https://dlmf.nist.gov/25.6.E19 25.6.E19] || <math qid="Q7656">\left(m+n+\tfrac{3}{2}\right)\Riemannzeta@{2m+2n+2} = \left(\sum_{k=1}^{m}+\sum_{k=1}^{n}\right)\Riemannzeta@{2k}\Riemannzeta@{2m+2n+2-2k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\left(m+n+\tfrac{3}{2}\right)\Riemannzeta@{2m+2n+2} = \left(\sum_{k=1}^{m}+\sum_{k=1}^{n}\right)\Riemannzeta@{2k}\Riemannzeta@{2m+2n+2-2k}</syntaxhighlight> || <math>m \geq 0, n \geq 0, m+n \geq 1</math> || <syntaxhighlight lang=mathematica>(m + n +(3)/(2))*Zeta(2*m + 2*n + 2) = (sum(, k = 1..m)+ sum(, k = 1..n))*Zeta(2*k)*Zeta(2*m + 2*n + 2 - 2*k)</syntaxhighlight> || <syntaxhighlight lang=mathematica>(m + n +Divide[3,2])*Zeta[2*m + 2*n + 2] == (Sum[, {k, 1, m}, GenerateConditions->None]+ Sum[, {k, 1, n}, GenerateConditions->None])*Zeta[2*k]*Zeta[2*m + 2*n + 2 - 2*k]</syntaxhighlight> || Translation Error || Translation Error || - || -
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| [https://dlmf.nist.gov/25.6.E20 25.6.E20] || <math qid="Q7657">\tfrac{1}{2}(2^{2n}-1)\Riemannzeta@{2n} = \sum_{k=1}^{n-1}(2^{2n-2k}-1)\Riemannzeta@{2n-2k}\Riemannzeta@{2k}</math><br><syntaxhighlight lang="tex" style="font-size: 75%;" inline>\tfrac{1}{2}(2^{2n}-1)\Riemannzeta@{2n} = \sum_{k=1}^{n-1}(2^{2n-2k}-1)\Riemannzeta@{2n-2k}\Riemannzeta@{2k}</syntaxhighlight> || <math>n \geq 2</math> || <syntaxhighlight lang=mathematica>(1)/(2)*((2)^(2*n)- 1)*Zeta(2*n) = sum(((2)^(2*n - 2*k)- 1)*Zeta(2*n - 2*k)*Zeta(2*k), k = 1..n - 1)</syntaxhighlight> || <syntaxhighlight lang=mathematica>Divide[1,2]*((2)^(2*n)- 1)*Zeta[2*n] == Sum[((2)^(2*n - 2*k)- 1)*Zeta[2*n - 2*k]*Zeta[2*k], {k, 1, n - 1}, GenerateConditions->None]</syntaxhighlight> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
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