25.2

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
25.2.E1	${\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{n=1}^{\infty}\frac{1}{n^{s}}}}$ \Riemannzeta@{s} = \sum_{n=1}^{\infty}\frac{1}{n^{s}}		Zeta(s) = sum((1)/((n)^(s)), n = 1..infinity)	Zeta[s] == Sum[Divide[1,(n)^(s)], {n, 1, Infinity}, GenerateConditions->None]	Failure	Successful	Failed [4 / 6] Result: Float(-infinity) Test Values: {s = -3/2} Result: Float(-infinity) Test Values: {s = -1/2} ... skip entries to safe data	Successful [Tested: 6]
25.2.E2	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}}}}$ \Riemannzeta@{s} = \frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}}	${\displaystyle{\displaystyle\Re s>1}}$	Zeta(s) = (1)/(1 - (2)^(- s))*sum((1)/((2*n + 1)^(s)), n = 0..infinity)	Zeta[s] == Divide[1,1 - (2)^(- s)]*Sum[Divide[1,(2*n + 1)^(s)], {n, 0, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 2]
25.2.E3	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}}}$ \Riemannzeta@{s} = \frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}	${\displaystyle{\displaystyle\Re s>0}}$	Zeta(s) = (1)/(1 - (2)^(1 - s))*sum(((- 1)^(n - 1))/((n)^(s)), n = 1..infinity)	Zeta[s] == Divide[1,1 - (2)^(1 - s)]*Sum[Divide[(- 1)^(n - 1),(n)^(s)], {n, 1, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
25.2.E4	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\gamma_{n}(s-1)^{n}}}$ \Riemannzeta@{s} = \frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\StieltjesConstants{n}(s-1)^{n}		Zeta(s) = (1)/(s - 1)+ sum(((- 1)^(n))/(factorial(n))*gamma(n)*(s - 1)^(n), n = 0..infinity)	Error	Failure	Missing Macro Error	Successful [Tested: 6]	-
25.2.E5	${\displaystyle{\displaystyle\gamma_{n}=\lim_{m\to\infty}\left(\sum_{k=1}^{m}\frac{(\ln k)^{n}}{k}-\frac{(\ln m)^{n+1}}{n+1}\right)}}$ \StieltjesConstants{n} = \lim_{m\to\infty}\left(\sum_{k=1}^{m}\frac{(\ln@@{k})^{n}}{k}-\frac{(\ln@@{m})^{n+1}}{n+1}\right)		gamma(n) = limit(sum(((ln(k))^(n))/(k), k = 1..m)-((ln(m))^(n + 1))/(n + 1), m = infinity)	Error	Successful	Missing Macro Error	-	-
25.2.E6	${\displaystyle{\displaystyle\zeta'\left(s\right)=-\sum_{n=2}^{\infty}(\ln n)n^{-s}}}$ \Riemannzeta'@{s} = -\sum_{n=2}^{\infty}(\ln@@{n})n^{-s}	${\displaystyle{\displaystyle\Re s>1}}$	diff( Zeta(s), s$(1) ) = - sum((ln(n))*(n)^(- s), n = 2..infinity)	D[Zeta[s], {s, 1}] == - Sum[(Log[n])*(n)^(- s), {n, 2, Infinity}, GenerateConditions->None]	Successful	Successful	-	Successful [Tested: 2]
25.2.E7	${\displaystyle{\displaystyle{\zeta^{(k)}}\left(s\right)=(-1)^{k}\sum_{n=2}^{\infty}(\ln n)^{k}n^{-s}}}$ \Riemannzeta^{(k)}@{s} = (-1)^{k}\sum_{n=2}^{\infty}(\ln@@{n})^{k}n^{-s}	${\displaystyle{\displaystyle\Re s>1}}$	diff( Zeta(s), s$(k) ) = (- 1)^(k)* sum((ln(n))^(k)* (n)^(- s), n = 2..infinity)	D[Zeta[s], {s, k}] == (- 1)^(k)* Sum[(Log[n])^(k)* (n)^(- s), {n, 2, Infinity}, GenerateConditions->None]	Failure	Failure	Successful [Tested: 2]	Successful [Tested: 2]
25.2.E8	${\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{s+1}}\mathrm{d}x}}$ \Riemannzeta@{s} = \sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\floor{x}}{x^{s+1}}\diff{x}	${\displaystyle{\displaystyle\Re s>0}}$	Zeta(s) = sum((1)/((k)^(s)), k = 1..N)+((N)^(1 - s))/(s - 1)- s*int((x - floor(x))/((x)^(s + 1)), x = N..infinity)	Zeta[s] == Sum[Divide[1,(k)^(s)], {k, 1, N}, GenerateConditions->None]+Divide[(N)^(1 - s),s - 1]- s*Integrate[Divide[x - Floor[x],(x)^(s + 1)], {x, N, Infinity}, GenerateConditions->None]	Failure	Aborted	Failed [3 / 3] Result: .2180864797 Test Values: {s = 3/2, N = 3} Result: Float(infinity) Test Values: {s = 1/2, N = 3} ... skip entries to safe data	Skipped - Because timed out
25.2.E11	${\displaystyle{\displaystyle\zeta\left(s\right)=\prod_{p}(1-p^{-s})^{-1}}}$ \Riemannzeta@{s} = \prod_{p}(1-p^{-s})^{-1}	${\displaystyle{\displaystyle\Re s>1}}$	Zeta(s) = product((1 - (p)^(- s))^(- 1), p = - infinity..infinity)	Zeta[s] == Product[(1 - (p)^(- s))^(- 1), {p, - Infinity, Infinity}, GenerateConditions->None]	Failure	Failure	Error	Failed [2 / 2] Result: Plus[2.612375348685488, Times[-1.0, NProduct[Power[Plus[1, Times[-1, Power[p, -1.5]]], -1] Test Values: {p, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, 1.5]} Result: Plus[1.6449340668482262, Times[-1.0, NProduct[Power[Plus[1, Times[-1, Power[p, -2]]], -1] Test Values: {p, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, 2]}
25.2.E12	${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{(2\pi)^{s}e^{-s-(\gamma s/2)}}{2(s-1)\Gamma\left(\tfrac{1}{2}s+1\right)}\prod_{\rho}\left(1-\frac{s}{\rho}\right)e^{s/\rho}}}$ \Riemannzeta@{s} = \frac{(2\pi)^{s}e^{-s-(\EulerConstant s/2)}}{2(s-1)\EulerGamma@{\tfrac{1}{2}s+1}}\prod_{\rho}\left(1-\frac{s}{\rho}\right)e^{s/\rho}	${\displaystyle{\displaystyle\Re(\tfrac{1}{2}s+1)>0}}$	Zeta(s) = ((2*Pi)^(s)* exp(- s -(gamma*s/2)))/(2*(s - 1)*GAMMA((1)/(2)*s + 1))*product((1 -(s)/(rho))*exp(s/rho), rho = - infinity..infinity)	Zeta[s] == Divide[(2*Pi)^(s)* Exp[- s -(EulerGamma*s/2)],2*(s - 1)*Gamma[Divide[1,2]*s + 1]]*Product[(1 -Divide[s,\[Rho]])*Exp[s/\[Rho]], {\[Rho], - Infinity, Infinity}, GenerateConditions->None]	Failure	Failure	Error	Failed [5 / 5] Result: Plus[Complex[-0.02548520188983307, 3.121037092000815*^-18], Times[0.02420092827533985, NProduct[Times[Power[E, Times[-1.5, Power[ρ, -1]]], Plus[1, Times[1.5, Power[ρ, -1]]]] Test Values: {ρ, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, -1.5]} Result: Plus[2.612375348685488, Times[-2.4801151038890965, NProduct[Times[Power[E, Times[1.5, Power[ρ, -1]]], Plus[1, Times[-1.5, Power[ρ, -1]]]] Test Values: {ρ, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, 1.5]} ... skip entries to safe data