Zeta and Related Functions - 25.6 Integer Arguments

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
25.6#Ex1 ζ ⁡ ( 0 ) = - 1 2 Riemann-zeta 0 1 2 {\displaystyle{\displaystyle\zeta\left(0\right)=-\frac{1}{2}}}
\Riemannzeta@{0} = -\frac{1}{2}		 

Zeta(0) = -(1)/(2)

Zeta[0] == -Divide[1,2]
Successful Successful - Successful [Tested: 1]
25.6#Ex2 ζ ⁡ ( 2 ) = π 2 6 Riemann-zeta 2 superscript 𝜋 2 6 {\displaystyle{\displaystyle\zeta\left(2\right)=\frac{\pi^{2}}{6}}}
\Riemannzeta@{2} = \frac{\pi^{2}}{6}		 

Zeta(2) = ((Pi)^(2))/(6)

Zeta[2] == Divide[(Pi)^(2),6]
Successful Successful - Successful [Tested: 1]
25.6#Ex3 ζ ⁡ ( 4 ) = π 4 90 Riemann-zeta 4 superscript 𝜋 4 90 {\displaystyle{\displaystyle\zeta\left(4\right)=\frac{\pi^{4}}{90}}}
\Riemannzeta@{4} = \frac{\pi^{4}}{90}		 

Zeta(4) = ((Pi)^(4))/(90)

Zeta[4] == Divide[(Pi)^(4),90]
Successful Successful - Successful [Tested: 1]
25.6#Ex4 ζ ⁡ ( 6 ) = π 6 945 Riemann-zeta 6 superscript 𝜋 6 945 {\displaystyle{\displaystyle\zeta\left(6\right)=\frac{\pi^{6}}{945}}}
\Riemannzeta@{6} = \frac{\pi^{6}}{945}		 

Zeta(6) = ((Pi)^(6))/(945)

Zeta[6] == Divide[(Pi)^(6),945]
Successful Successful - Successful [Tested: 1]
25.6.E2 ζ ⁡ ( 2 ⁢ n ) = ( 2 ⁢ π ) 2 ⁢ n 2 ⁢ ( 2 ⁢ n ) ! ⁢ | B 2 ⁢ n | Riemann-zeta 2 𝑛 superscript 2 𝜋 2 𝑛 2 2 𝑛 Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle\zeta\left(2n\right)=\frac{(2\pi)^{2n}}{2(2n)!}% \left|B_{2n}\right|}}
\Riemannzeta@{2n} = \frac{(2\pi)^{2n}}{2(2n)!}\left|\BernoullinumberB{2n}\right|		 

Zeta(2*n) = ((2*Pi)^(2*n))/(2*factorial(2*n))*abs(bernoulli(2*n))

Zeta[2*n] == Divide[(2*Pi)^(2*n),2*(2*n)!]*Abs[BernoulliB[2*n]]
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
25.6.E3 ζ ⁡ ( - n ) = - B n + 1 n + 1 Riemann-zeta 𝑛 Bernoulli-number-B 𝑛 1 𝑛 1 {\displaystyle{\displaystyle\zeta\left(-n\right)=-\frac{B_{n+1}}{n+1}}}
\Riemannzeta@{-n} = -\frac{\BernoullinumberB{n+1}}{n+1}		 

Zeta(- n) = -(bernoulli(n + 1))/(n + 1)

Zeta[- n] == -Divide[BernoulliB[n + 1],n + 1]
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
25.6.E4 ζ ⁡ ( - 2 ⁢ n ) = 0 Riemann-zeta 2 𝑛 0 {\displaystyle{\displaystyle\zeta\left(-2n\right)=0}}
\Riemannzeta@{-2n} = 0		 

Zeta(- 2*n) = 0

Zeta[- 2*n] == 0
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
25.6.E6 ζ ⁡ ( 2 ⁢ k + 1 ) = ( - 1 ) k + 1 ⁢ ( 2 ⁢ π ) 2 ⁢ k + 1 2 ⁢ ( 2 ⁢ k + 1 ) ! ⁢ ∫ 0 1 B 2 ⁢ k + 1 ⁡ ( t ) ⁢ cot ⁡ ( π ⁢ t ) ⁢ d t Riemann-zeta 2 𝑘 1 superscript 1 𝑘 1 superscript 2 𝜋 2 𝑘 1 2 2 𝑘 1 superscript subscript 0 1 Bernoulli-polynomial-B 2 𝑘 1 𝑡 𝜋 𝑡 𝑡 {\displaystyle{\displaystyle\zeta\left(2k+1\right)=\frac{(-1)^{k+1}(2\pi)^{2k+% 1}}{2(2k+1)!}\int_{0}^{1}B_{2k+1}\left(t\right)\cot\left(\pi t\right)\mathrm{d% }t}}
\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}		 

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)

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]
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
25.6.E7 ζ ⁡ ( 2 ) = ∫ 0 1 ∫ 0 1 1 1 - x ⁢ y ⁢ d x ⁢ d y Riemann-zeta 2 superscript subscript 0 1 superscript subscript 0 1 1 1 𝑥 𝑦 𝑥 𝑦 {\displaystyle{\displaystyle\zeta\left(2\right)=\int_{0}^{1}\int_{0}^{1}\frac{% 1}{1-xy}\mathrm{d}x\mathrm{d}y}}
\Riemannzeta@{2} = \int_{0}^{1}\int_{0}^{1}\frac{1}{1-xy}\diff{x}\diff{y}		 

Zeta(2) = int(int((1)/(1 - x*y), x = 0..1), y = 0..1)

Zeta[2] == Integrate[Integrate[Divide[1,1 - x*y], {x, 0, 1}, GenerateConditions->None], {y, 0, 1}, GenerateConditions->None]
Successful Successful - Successful [Tested: 1]
25.6.E8 ζ ⁡ ( 2 ) = 3 ⁢ ∑ k = 1 ∞ 1 k 2 ⁢ ( 2 ⁢ k k ) Riemann-zeta 2 3 superscript subscript 𝑘 1 1 superscript 𝑘 2 binomial 2 𝑘 𝑘 {\displaystyle{\displaystyle\zeta\left(2\right)=3\sum_{k=1}^{\infty}\frac{1}{k% ^{2}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}
\Riemannzeta@{2} = 3\sum_{k=1}^{\infty}\frac{1}{k^{2}\binom{2k}{k}}		 

Zeta(2) = 3*sum((1)/((k)^(2)*binomial(2*k,k)), k = 1..infinity)

Zeta[2] == 3*Sum[Divide[1,(k)^(2)*Binomial[2*k,k]], {k, 1, Infinity}, GenerateConditions->None]
Successful Successful - Successful [Tested: 1]
25.6.E9 ζ ⁡ ( 3 ) = 5 2 ⁢ ∑ k = 1 ∞ ( - 1 ) k - 1 k 3 ⁢ ( 2 ⁢ k k ) Riemann-zeta 3 5 2 superscript subscript 𝑘 1 superscript 1 𝑘 1 superscript 𝑘 3 binomial 2 𝑘 𝑘 {\displaystyle{\displaystyle\zeta\left(3\right)=\frac{5}{2}\sum_{k=1}^{\infty}% \frac{(-1)^{k-1}}{k^{3}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}
\Riemannzeta@{3} = \frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^{3}\binom{2k}{k}}		 

Zeta(3) = (5)/(2)*sum(((- 1)^(k - 1))/((k)^(3)*binomial(2*k,k)), k = 1..infinity)

Zeta[3] == Divide[5,2]*Sum[Divide[(- 1)^(k - 1),(k)^(3)*Binomial[2*k,k]], {k, 1, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
25.6.E10 ζ ⁡ ( 4 ) = 36 17 ⁢ ∑ k = 1 ∞ 1 k 4 ⁢ ( 2 ⁢ k k ) Riemann-zeta 4 36 17 superscript subscript 𝑘 1 1 superscript 𝑘 4 binomial 2 𝑘 𝑘 {\displaystyle{\displaystyle\zeta\left(4\right)=\frac{36}{17}\sum_{k=1}^{% \infty}\frac{1}{k^{4}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}
\Riemannzeta@{4} = \frac{36}{17}\sum_{k=1}^{\infty}\frac{1}{k^{4}\binom{2k}{k}}		 

Zeta(4) = (36)/(17)*sum((1)/((k)^(4)*binomial(2*k,k)), k = 1..infinity)

Zeta[4] == Divide[36,17]*Sum[Divide[1,(k)^(4)*Binomial[2*k,k]], {k, 1, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 0] Successful [Tested: 1]
25.6.E11 ζ ′ ⁡ ( 0 ) = - 1 2 ⁢ ln ⁡ ( 2 ⁢ π ) diffop Riemann-zeta 1 0 1 2 2 𝜋 {\displaystyle{\displaystyle\zeta'\left(0\right)=-\tfrac{1}{2}\ln\left(2\pi% \right)}}
\Riemannzeta'@{0} = -\tfrac{1}{2}\ln@{2\pi}		 

subs( temp=0, diff( Zeta(temp), temp$(1) ) ) = -(1)/(2)*ln(2*Pi)

(D[Zeta[temp], {temp, 1}]/.temp-> 0) == -Divide[1,2]*Log[2*Pi]
Successful Successful - Successful [Tested: 1]
25.6.E12 ζ ′′ ⁡ ( 0 ) = - 1 2 ⁢ ( ln ⁡ ( 2 ⁢ π ) ) 2 + 1 2 ⁢ γ 2 - 1 24 ⁢ π 2 + γ 1 diffop Riemann-zeta 2 0 1 2 superscript 2 𝜋 2 1 2 2 1 24 superscript 𝜋 2 Stieltjes-constants 1 {\displaystyle{\displaystyle\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi% \right))^{2}+\tfrac{1}{2}{\gamma^{2}}-\tfrac{1}{24}\pi^{2}+\gamma_{1}}}
\Riemannzeta''@{0} = -\tfrac{1}{2}(\ln@{2\pi})^{2}+\tfrac{1}{2}\EulerConstant^{2}-\tfrac{1}{24}\pi^{2}+\StieltjesConstants{1}		 

subs( temp=0, diff( Zeta(temp), temp$(2) ) ) = -(1)/(2)*(ln(2*Pi))^(2)+(1)/(2)*(gamma)^(2)-(1)/(24)*(Pi)^(2)+ gamma(1)

Error
Successful Missing Macro Error - -
25.6.E13 ( - 1 ) k ⁢ ζ ( k ) ⁡ ( - 2 ⁢ n ) = 2 ⁢ ( - 1 ) n ( 2 ⁢ π ) 2 ⁢ n + 1 ⁢ ∑ m = 0 k ∑ r = 0 m ( k m ) ⁢ ( m r ) ⁢ ℑ ⁡ ( c k - m ) ⁢ Γ ( r ) ⁡ ( 2 ⁢ n + 1 ) ⁢ ζ ( m - r ) ⁡ ( 2 ⁢ n + 1 ) superscript 1 𝑘 Riemann-zeta 𝑘 2 𝑛 2 superscript 1 𝑛 superscript 2 𝜋 2 𝑛 1 superscript subscript 𝑚 0 𝑘 superscript subscript 𝑟 0 𝑚 binomial 𝑘 𝑚 binomial 𝑚 𝑟 superscript 𝑐 𝑘 𝑚 Euler-Gamma 𝑟 2 𝑛 1 Riemann-zeta 𝑚 𝑟 2 𝑛 1 {\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(-2n\right)=\frac{2(-1)^% {n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}% \genfrac{(}{)}{0.0pt}{}{m}{r}\Im\left(c^{k-m}\right)\*{\Gamma^{(r)}}\left(2n+1% \right){\zeta^{(m-r)}}\left(2n+1\right)}}
(-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}		 ℜ ⁡ ( 2 ⁢ n + 1 ) > 0 2 𝑛 1 0 {\displaystyle{\displaystyle\Re(2n+1)>0}} 
(- 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)

(- 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]
Aborted Failure Skipped - Because timed out
Failed [48 / 54]
Result: 0.030448457058393275
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[n, 1]}


Result: -0.007983811450268627
Test Values: {Rule[c, -1.5], Rule[k, 1], Rule[n, 2]}

... skip entries to safe data
25.6.E14 ( - 1 ) k ⁢ ζ ( k ) ⁡ ( 1 - 2 ⁢ n ) = 2 ⁢ ( - 1 ) n ( 2 ⁢ π ) 2 ⁢ n ⁢ ∑ m = 0 k ∑ r = 0 m ( k m ) ⁢ ( m r ) ⁢ ℜ ⁡ ( c k - m ) ⁢ Γ ( r ) ⁡ ( 2 ⁢ n ) ⁢ ζ ( m - r ) ⁡ ( 2 ⁢ n ) superscript 1 𝑘 Riemann-zeta 𝑘 1 2 𝑛 2 superscript 1 𝑛 superscript 2 𝜋 2 𝑛 superscript subscript 𝑚 0 𝑘 superscript subscript 𝑟 0 𝑚 binomial 𝑘 𝑚 binomial 𝑚 𝑟 superscript 𝑐 𝑘 𝑚 Euler-Gamma 𝑟 2 𝑛 Riemann-zeta 𝑚 𝑟 2 𝑛 {\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(1-2n\right)=\frac{2(-1)% ^{n}}{(2\pi)^{2n}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}% \genfrac{(}{)}{0.0pt}{}{m}{r}\Re\left(c^{k-m}\right)\*{\Gamma^{(r)}}\left(2n% \right){\zeta^{(m-r)}}\left(2n\right)}}
(-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}		 ℜ ⁡ ( 2 ⁢ n ) > 0 2 𝑛 0 {\displaystyle{\displaystyle\Re(2n)>0}} 
(- 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)

(- 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]
Aborted Failure Skipped - Because timed out
Failed [54 / 54]
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]}


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]}

... skip entries to safe data
25.6.E15 ζ ′ ⁡ ( 2 ⁢ n ) = ( - 1 ) n + 1 ⁢ ( 2 ⁢ π ) 2 ⁢ n 2 ⁢ ( 2 ⁢ n ) ! ⁢ ( 2 ⁢ n ⁢ ζ ′ ⁡ ( 1 - 2 ⁢ n ) - ( ψ ⁡ ( 2 ⁢ n ) - ln ⁡ ( 2 ⁢ π ) ) ⁢ B 2 ⁢ n ) diffop Riemann-zeta 1 2 𝑛 superscript 1 𝑛 1 superscript 2 𝜋 2 𝑛 2 2 𝑛 2 𝑛 diffop Riemann-zeta 1 1 2 𝑛 digamma 2 𝑛 2 𝜋 Bernoulli-number-B 2 𝑛 {\displaystyle{\displaystyle\zeta'\left(2n\right)=\frac{(-1)^{n+1}(2\pi)^{2n}}% {2(2n)!}\left(2n\zeta'\left(1-2n\right)-(\psi\left(2n\right)-\ln\left(2\pi% \right))B_{2n}\right)}}
\Riemannzeta'@{2n} = \frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\Riemannzeta'@{1-2n}-(\digamma@{2n}-\ln@{2\pi})\BernoullinumberB{2n}\right)		 

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))

(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])
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
25.6.E16 ( n + 1 2 ) ⁢ ζ ⁡ ( 2 ⁢ n ) = ∑ k = 1 n - 1 ζ ⁡ ( 2 ⁢ k ) ⁢ ζ ⁡ ( 2 ⁢ n - 2 ⁢ k ) 𝑛 1 2 Riemann-zeta 2 𝑛 superscript subscript 𝑘 1 𝑛 1 Riemann-zeta 2 𝑘 Riemann-zeta 2 𝑛 2 𝑘 {\displaystyle{\displaystyle\left(n+\tfrac{1}{2}\right)\zeta\left(2n\right)=% \sum_{k=1}^{n-1}\zeta\left(2k\right)\zeta\left(2n-2k\right)}}
\left(n+\tfrac{1}{2}\right)\Riemannzeta@{2n} = \sum_{k=1}^{n-1}\Riemannzeta@{2k}\Riemannzeta@{2n-2k}		 n ≥ 2 𝑛 2 {\displaystyle{\displaystyle n\geq 2}} 
(n +(1)/(2))*Zeta(2*n) = sum(Zeta(2*k)*Zeta(2*n - 2*k), k = 1..n - 1)

(n +Divide[1,2])*Zeta[2*n] == Sum[Zeta[2*k]*Zeta[2*n - 2*k], {k, 1, n - 1}, GenerateConditions->None]
Failure Failure Successful [Tested: 2] Successful [Tested: 2]
25.6.E17 ( n + 3 4 ) ⁢ ζ ⁡ ( 4 ⁢ n + 2 ) = ∑ k = 1 n ζ ⁡ ( 2 ⁢ k ) ⁢ ζ ⁡ ( 4 ⁢ n + 2 - 2 ⁢ k ) 𝑛 3 4 Riemann-zeta 4 𝑛 2 superscript subscript 𝑘 1 𝑛 Riemann-zeta 2 𝑘 Riemann-zeta 4 𝑛 2 2 𝑘 {\displaystyle{\displaystyle\left(n+\tfrac{3}{4}\right)\zeta\left(4n+2\right)=% \sum_{k=1}^{n}\zeta\left(2k\right)\zeta\left(4n+2-2k\right)}}
\left(n+\tfrac{3}{4}\right)\Riemannzeta@{4n+2} = \sum_{k=1}^{n}\Riemannzeta@{2k}\Riemannzeta@{4n+2-2k}		 n ≥ 1 𝑛 1 {\displaystyle{\displaystyle n\geq 1}} 
(n +(3)/(4))*Zeta(4*n + 2) = sum(Zeta(2*k)*Zeta(4*n + 2 - 2*k), k = 1..n)

(n +Divide[3,4])*Zeta[4*n + 2] == Sum[Zeta[2*k]*Zeta[4*n + 2 - 2*k], {k, 1, n}, GenerateConditions->None]
Failure Failure Successful [Tested: 3] Successful [Tested: 3]
25.6.E18 ( n + 1 4 ) ⁢ ζ ⁡ ( 4 ⁢ n ) + 1 2 ⁢ ( ζ ⁡ ( 2 ⁢ n ) ) 2 = ∑ k = 1 n ζ ⁡ ( 2 ⁢ k ) ⁢ ζ ⁡ ( 4 ⁢ n - 2 ⁢ k ) 𝑛 1 4 Riemann-zeta 4 𝑛 1 2 superscript Riemann-zeta 2 𝑛 2 superscript subscript 𝑘 1 𝑛 Riemann-zeta 2 𝑘 Riemann-zeta 4 𝑛 2 𝑘 {\displaystyle{\displaystyle\left(n+\tfrac{1}{4}\right)\zeta\left(4n\right)+% \tfrac{1}{2}(\zeta\left(2n\right))^{2}=\sum_{k=1}^{n}\zeta\left(2k\right)\zeta% \left(4n-2k\right)}}
\left(n+\tfrac{1}{4}\right)\Riemannzeta@{4n}+\tfrac{1}{2}(\Riemannzeta@{2n})^{2} = \sum_{k=1}^{n}\Riemannzeta@{2k}\Riemannzeta@{4n-2k}		 n ≥ 1 𝑛 1 {\displaystyle{\displaystyle n\geq 1}} 
(n +(1)/(4))*Zeta(4*n)+(1)/(2)*(Zeta(2*n))^(2) = sum(Zeta(2*k)*Zeta(4*n - 2*k), k = 1..n)

(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]
Skipped - Unable to analyze test case: Null Skipped - Unable to analyze test case: Null - -
25.6.E19 ( m + n + 3 2 ) ⁢ ζ ⁡ ( 2 ⁢ m + 2 ⁢ n + 2 ) = ( ∑ k = 1 m + ∑ k = 1 n ) ⁢ ζ ⁡ ( 2 ⁢ k ) ⁢ ζ ⁡ ( 2 ⁢ m + 2 ⁢ n + 2 - 2 ⁢ k ) 𝑚 𝑛 3 2 Riemann-zeta 2 𝑚 2 𝑛 2 superscript subscript 𝑘 1 𝑚 superscript subscript 𝑘 1 𝑛 Riemann-zeta 2 𝑘 Riemann-zeta 2 𝑚 2 𝑛 2 2 𝑘 {\displaystyle{\displaystyle\left(m+n+\tfrac{3}{2}\right)\zeta\left(2m+2n+2% \right)=\left(\sum_{k=1}^{m}+\sum_{k=1}^{n}\right)\zeta\left(2k\right)\zeta% \left(2m+2n+2-2k\right)}}
\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}		 m ≥ 0 , n ≥ 0 , m + n ≥ 1 formulae-sequence 𝑚 0 formulae-sequence 𝑛 0 𝑚 𝑛 1 {\displaystyle{\displaystyle m\geq 0,n\geq 0,m+n\geq 1}} 
(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)

(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]
Translation Error Translation Error - -
25.6.E20 1 2 ⁢ ( 2 2 ⁢ n - 1 ) ⁢ ζ ⁡ ( 2 ⁢ n ) = ∑ k = 1 n - 1 ( 2 2 ⁢ n - 2 ⁢ k - 1 ) ⁢ ζ ⁡ ( 2 ⁢ n - 2 ⁢ k ) ⁢ ζ ⁡ ( 2 ⁢ k ) 1 2 superscript 2 2 𝑛 1 Riemann-zeta 2 𝑛 superscript subscript 𝑘 1 𝑛 1 superscript 2 2 𝑛 2 𝑘 1 Riemann-zeta 2 𝑛 2 𝑘 Riemann-zeta 2 𝑘 {\displaystyle{\displaystyle\tfrac{1}{2}(2^{2n}-1)\zeta\left(2n\right)=\sum_{k% =1}^{n-1}(2^{2n-2k}-1)\zeta\left(2n-2k\right)\zeta\left(2k\right)}}
\tfrac{1}{2}(2^{2n}-1)\Riemannzeta@{2n} = \sum_{k=1}^{n-1}(2^{2n-2k}-1)\Riemannzeta@{2n-2k}\Riemannzeta@{2k}		 n ≥ 2 𝑛 2 {\displaystyle{\displaystyle n\geq 2}} 
(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)

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]
Failure Failure Successful [Tested: 2] Successful [Tested: 2]
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