25.6

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
25.6#Ex1	${\displaystyle{\displaystyle\zeta\left(0\right)=-\frac{1}{2}}}$ \Riemannzeta@{0} = -\frac{1}{2}		Zeta(0) = -(1)/(2)	Zeta[0] == -Divide[1,2]	Successful	Successful	-	Successful [Tested: 1]
25.6#Ex2	${\displaystyle{\displaystyle\zeta\left(2\right)=\frac{\pi^{2}}{6}}}$ \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	${\displaystyle{\displaystyle\zeta\left(4\right)=\frac{\pi^{4}}{90}}}$ \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	${\displaystyle{\displaystyle\zeta\left(6\right)=\frac{\pi^{6}}{945}}}$ \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	${\displaystyle{\displaystyle\zeta\left(2n\right)=\frac{(2\pi)^{2n}}{2(2n)!}\left|B_{2n}\right|}}$ \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	${\displaystyle{\displaystyle\zeta\left(-n\right)=-\frac{B_{n+1}}{n+1}}}$ \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	${\displaystyle{\displaystyle\zeta\left(-2n\right)=0}}$ \Riemannzeta@{-2n} = 0		Zeta(- 2*n) = 0	Zeta[- 2*n] == 0	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
25.6.E6	${\displaystyle{\displaystyle\zeta\left(2k+1\right)=\frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}B_{2k+1}\left(t\right)\cot\left(\pi t\right)\mathrm{d}t}}$ \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	${\displaystyle{\displaystyle\zeta\left(2\right)=\int_{0}^{1}\int_{0}^{1}\frac{1}{1-xy}\mathrm{d}x\mathrm{d}y}}$ \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	${\displaystyle{\displaystyle\zeta\left(2\right)=3\sum_{k=1}^{\infty}\frac{1}{k^{2}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}$ \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	${\displaystyle{\displaystyle\zeta\left(3\right)=\frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^{3}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}$ \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	${\displaystyle{\displaystyle\zeta\left(4\right)=\frac{36}{17}\sum_{k=1}^{\infty}\frac{1}{k^{4}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}$ \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	${\displaystyle{\displaystyle\zeta'\left(0\right)=-\tfrac{1}{2}\ln\left(2\pi\right)}}$ \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	${\displaystyle{\displaystyle\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi\right))^{2}+\tfrac{1}{2}{\gamma^{2}}-\tfrac{1}{24}\pi^{2}+\gamma_{1}}}$ \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	${\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(-2n\right)=\frac{2(-1)^{n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\Im\left(c^{k-m}\right)\*{\Gamma^{(r)}}\left(2n+1\right){\zeta^{(m-r)}}\left(2n+1\right)}}$ (-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}	${\displaystyle{\displaystyle\Re(2n+1)>0}}$	(- 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	${\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(1-2n\right)=\frac{2(-1)^{n}}{(2\pi)^{2n}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{)}{0.0pt}{}{m}{r}\Re\left(c^{k-m}\right)\*{\Gamma^{(r)}}\left(2n\right){\zeta^{(m-r)}}\left(2n\right)}}$ (-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}	${\displaystyle{\displaystyle\Re(2n)>0}}$	(- 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	${\displaystyle{\displaystyle\zeta'\left(2n\right)=\frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\zeta'\left(1-2n\right)-(\psi\left(2n\right)-\ln\left(2\pi\right))B_{2n}\right)}}$ \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	${\displaystyle{\displaystyle\left(n+\tfrac{1}{2}\right)\zeta\left(2n\right)=\sum_{k=1}^{n-1}\zeta\left(2k\right)\zeta\left(2n-2k\right)}}$ \left(n+\tfrac{1}{2}\right)\Riemannzeta@{2n} = \sum_{k=1}^{n-1}\Riemannzeta@{2k}\Riemannzeta@{2n-2k}	${\displaystyle{\displaystyle n\geq 2}}$	(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	${\displaystyle{\displaystyle\left(n+\tfrac{3}{4}\right)\zeta\left(4n+2\right)=\sum_{k=1}^{n}\zeta\left(2k\right)\zeta\left(4n+2-2k\right)}}$ \left(n+\tfrac{3}{4}\right)\Riemannzeta@{4n+2} = \sum_{k=1}^{n}\Riemannzeta@{2k}\Riemannzeta@{4n+2-2k}	${\displaystyle{\displaystyle n\geq 1}}$	(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	${\displaystyle{\displaystyle\left(n+\tfrac{1}{4}\right)\zeta\left(4n\right)+\tfrac{1}{2}(\zeta\left(2n\right))^{2}=\sum_{k=1}^{n}\zeta\left(2k\right)\zeta\left(4n-2k\right)}}$ \left(n+\tfrac{1}{4}\right)\Riemannzeta@{4n}+\tfrac{1}{2}(\Riemannzeta@{2n})^{2} = \sum_{k=1}^{n}\Riemannzeta@{2k}\Riemannzeta@{4n-2k}	${\displaystyle{\displaystyle n\geq 1}}$	(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	${\displaystyle{\displaystyle\left(m+n+\tfrac{3}{2}\right)\zeta\left(2m+2n+2\right)=\left(\sum_{k=1}^{m}+\sum_{k=1}^{n}\right)\zeta\left(2k\right)\zeta\left(2m+2n+2-2k\right)}}$ \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}	${\displaystyle{\displaystyle m\geq 0,n\geq 0,m+n\geq 1}}$	(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	${\displaystyle{\displaystyle\tfrac{1}{2}(2^{2n}-1)\zeta\left(2n\right)=\sum_{k=1}^{n-1}(2^{2n-2k}-1)\zeta\left(2n-2k\right)\zeta\left(2k\right)}}$ \tfrac{1}{2}(2^{2n}-1)\Riemannzeta@{2n} = \sum_{k=1}^{n-1}(2^{2n-2k}-1)\Riemannzeta@{2n-2k}\Riemannzeta@{2k}	${\displaystyle{\displaystyle n\geq 2}}$	(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]