Functions of Number Theory - 27.4 Euler Products and Dirichlet Series

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
27.4.E3	${\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{n=1}^{\infty}n^{-s}}}$ \Riemannzeta@{s} = \sum_{n=1}^{\infty}n^{-s}	${\displaystyle{\displaystyle\Re s>1}}$	Zeta(s) = sum((n)^(- s), n = 1..infinity)	Zeta[s] == Sum[(n)^(- s), {n, 1, Infinity}, GenerateConditions->None]	Successful	Successful	Skip - symbolical successful subtest	Successful [Tested: 2]
27.4.E3	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}n^{-s}=\prod_{p}(1-p^{-s})^{-1}}}$ \sum_{n=1}^{\infty}n^{-s} = \prod_{p}(1-p^{-s})^{-1}	${\displaystyle{\displaystyle\Re s>1}}$	sum((n)^(- s), n = 1..infinity) = product((1 - (p)^(- s))^(- 1), p = - infinity..infinity)	Sum[(n)^(- s), {n, 1, Infinity}, GenerateConditions->None] == 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]}
27.4.E6	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}\phi\left(n\right)n^{-s}=\frac{\zeta\left(s-1\right)}{\zeta\left(s\right)}}}$ \sum_{n=1}^{\infty}\Eulertotientphi[]@{n}n^{-s} = \frac{\Riemannzeta@{s-1}}{\Riemannzeta@{s}}	${\displaystyle{\displaystyle\Re s>2}}$	sum(phi(n)*(n)^(- s), n = 1..infinity) = (Zeta(s - 1))/(Zeta(s))	Sum[EulerPhi[n]*(n)^(- s), {n, 1, Infinity}, GenerateConditions->None] == Divide[Zeta[s - 1],Zeta[s]]	Failure	Successful	Error	Successful [Tested: 0]
27.4.E9	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}2^{\nu\left(n\right)}n^{-s}=\frac{(\zeta\left(s\right))^{2}}{\zeta\left(2s\right)}}}$ \sum_{n=1}^{\infty}2^{\nprimesdiv@{n}}n^{-s} = \frac{(\Riemannzeta@{s})^{2}}{\Riemannzeta@{2s}}	${\displaystyle{\displaystyle\Re s>1}}$	sum((2)^(ifactor(n))* (n)^(- s), n = 1..infinity) = ((Zeta(s))^(2))/(Zeta(2*s))	Error	Error	Missing Macro Error	-	-
27.4.E11	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)n^{-s}=\zeta\left(s\right)\zeta\left(s-\alpha\right)}}$ \sum_{n=1}^{\infty}\sumdivisors{\alpha}@{n}n^{-s} = \Riemannzeta@{s}\Riemannzeta@{s-\alpha}	${\displaystyle{\displaystyle\Re s>\max(1}}$	sum(add(divisors(alpha))*(n)^(- s), n = 1..infinity) = Zeta(s)*Zeta(s - alpha)	Error	Failure	Missing Macro Error	Failed [18 / 18] Result: Float(infinity) Test Values: {alpha = 3/2, s = -3/2} Result: 5.224750698 Test Values: {alpha = 3/2, s = 3/2} ... skip entries to safe data	-
27.4.E13	${\displaystyle{\displaystyle\sum_{n=2}^{\infty}(\ln n)n^{-s}=-\zeta'\left(s\right)}}$ \sum_{n=2}^{\infty}(\ln@@{n})n^{-s} = -\Riemannzeta'@{s}	${\displaystyle{\displaystyle\Re s>1}}$	sum((ln(n))*(n)^(- s), n = 2..infinity) = - diff( Zeta(s), s$(1) )	Sum[(Log[n])*(n)^(- s), {n, 2, Infinity}, GenerateConditions->None] == - D[Zeta[s], {s, 1}]	Successful	Successful	-	Successful [Tested: 2]