Zeta and Related Functions - 25.16 Mathematical Applications

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
25.16.E10	${\displaystyle{\displaystyle\frac{1}{2}\zeta\left(1-2a\right)=-\frac{B_{2a}}{4a}}}$ \frac{1}{2}\Riemannzeta@{1-2a} = -\frac{\BernoullinumberB{2a}}{4a}		(1)/(2)*Zeta(1 - 2*a) = -(bernoulli(2*a))/(4*a)	Divide[1,2]*Zeta[1 - 2*a] == -Divide[BernoulliB[2*a],4*a]	Failure	Failure	Successful [Tested: 1]	Successful [Tested: 1]
25.16.E13	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}\left(\frac{h(n)}{n}\right)^{2}=\frac{17}{4}\zeta\left(4\right)}}$ \sum_{n=1}^{\infty}\left(\frac{h(n)}{n}\right)^{2} = \frac{17}{4}\Riemannzeta@{4}		sum(((h(n))/(n))^(2), n = 1..infinity) = (17)/(4)*Zeta(4)	Sum[(Divide[h[n],n])^(2), {n, 1, Infinity}, GenerateConditions->None] == Divide[17,4]*Zeta[4]	Failure	Failure	Failed [10 / 10] Result: Float(infinity)+Float(infinity)*I Test Values: {h = 1/2*3^(1/2)+1/2*I} Result: Float(infinity)+Float(infinity)*I Test Values: {h = -1/2+1/2*I*3^(1/2)} ... skip entries to safe data	Failed [10 / 10] Result: Plus[-4.599873743272337, NSum[Power[E, Times[Complex[0, Rational[1, 3]], Pi]] Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]} Result: Plus[-4.599873743272337, NSum[Power[E, Times[Complex[0, Rational[-2, 3]], Pi]] Test Values: {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[h, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]} ... skip entries to safe data
25.16.E14	${\displaystyle{\displaystyle\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)}=\frac{5}{4}\zeta\left(3\right)}}$ \sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)} = \frac{5}{4}\Riemannzeta@{3}		sum(sum((1)/(r*k*(r + k)), k = 1..r), r = 1..infinity) = (5)/(4)*Zeta(3)	Sum[Sum[Divide[1,r*k*(r + k)], {k, 1, r}, GenerateConditions->None], {r, 1, Infinity}, GenerateConditions->None] == Divide[5,4]*Zeta[3]	Failure	Aborted	Error	Successful [Tested: 1]