Results of Functions of Number Theory

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DLMF Formula Maple Mathematica Symbolic
Maple		 Symbolic
Mathematica		 Numeric
Maple		 Numeric
Mathematica
27.2.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n = \prod_{r=1}^{\nprimesdiv@{n}}p^{a_{r}}_{r}} n product(p(p[r])^(a[r]), r = 1..ifactor(n)) Error Error Translation Error - -
27.2.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Eulertotientphi[]@{n} = \Eulertotientphi[0]@{n}} Error EulerPhi[n] == Sum[If[CoprimeQ[n, m], m^(0), 0], {m, 1, n}] Missing Macro Error Failure -
Failed [3 / 3]
{1.0 <- {Rule[n, 1]}
1.0 <- {Rule[n, 2]}
27.2.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ndivisors[]@{n} = \sum_{d\divides n}1} numelems(Divisors(n)) = sum(1, d**n in - infinity) Error Translation Error Missing Macro Error - -
27.2.E10 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sumdivisors{\alpha}@{n} = \sum_{d\divides n}d^{\alpha}} add(divisors(alpha)) = sum((d)^(alpha), d**n in - infinity) Error Translation Error Missing Macro Error - -
27.3.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Eulertotientphi[]@{n} = n\prod_{p\divides n}(1-p^{-1})} phi(n) = n*product(1 - (p)^(- 1), p**n in - infinity) EulerPhi[n] == n*Product[1 - (p)^(- 1), {p**n, - Infinity}, GenerateConditions->None] Translation Error Translation Error - -
27.3.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \ndivisors[]@{n} = \prod_{r=1}^{\nprimesdiv@{n}}(1+a_{r})} numelems(Divisors(n)) = product(1 + a[r], r = 1..ifactor(n)) Error Error Missing Macro Error - -
27.3.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sumdivisors{\alpha}@{n} = \prod_{r=1}^{\nprimesdiv@{n}}\frac{p^{\alpha(1+a_{r})}_{r}-1}{p^{\alpha}_{r}-1}} product((p(p[r])^(alpha*(1 + a[r]))- 1)/(p(p[r])^(alpha)- 1), r = 1..ifactor(n)) Error Failure Missing Macro Error Error -
27.3.E8 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Eulertotientphi[]@{m}\Eulertotientphi[]@{n} = \Eulertotientphi[]@{mn}\Eulertotientphi[]@{\pgcd{m,n}}/\pgcd{m,n}} phi(m)*phi(n) = phi(m*n)*phi(gcd(m , n))/ gcd(m , n) EulerPhi[m]*EulerPhi[n] == EulerPhi[m*n]*EulerPhi[GCD[m , n]]/ GCD[m , n] Failure Failure
Failed [2 / 9]
2/9]: [[-1. <- {m = 2, n = 2}
-2. <- {m = 3, n = 3}
 Successful [Tested: 9]
27.4.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Riemannzeta@{s} = \sum_{n=1}^{\infty}n^{-s}} 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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}n^{-s} = \prod_{p}(1-p^{-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]
{Plus[2.612375348685488, Times[-1.0, NProduct[Power[Plus[1, Times[-1, Power[p, -1.5]]], -1] <- {p, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, 1.5]}
Plus[1.6449340668482262, Times[-1.0, NProduct[Power[Plus[1, Times[-1, Power[p, -2]]], -1] <- {p, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, 2]}
27.4.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}\Eulertotientphi[]@{n}n^{-s} = \frac{\Riemannzeta@{s-1}}{\Riemannzeta@{s}}} 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 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}2^{\nprimesdiv@{n}}n^{-s} = \frac{(\Riemannzeta@{s})^{2}}{\Riemannzeta@{2s}}} sum((2)^(ifactor(n))* (n)^(- s), n = 1..infinity) = ((Zeta(s))^(2))/(Zeta(2*s)) Error Error Missing Macro Error - -
27.4.E11 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}\sumdivisors{\alpha}@{n}n^{-s} = \Riemannzeta@{s}\Riemannzeta@{s-\alpha}} sum(add(divisors(alpha))*(n)^(- s), n = 1..infinity) = Zeta(s)*Zeta(s - alpha) Error Failure Missing Macro Error
Failed [18 / 18]
18/18]: [[Float(infinity) <- {alpha = 3/2, s = -3/2}
5.224750698 <- {alpha = 3/2, s = 3/2}
 -
27.4.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=2}^{\infty}(\ln@@{n})n^{-s} = -\Riemannzeta'@{s}} 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]
27.7.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}\Eulertotientphi[]@{n}\frac{x^{n}}{1-x^{n}} = \frac{x}{(1-x)^{2}}} sum(phi(n)*((x)^(n))/(1 - (x)^(n)), n = 1..infinity) = (x)/((1 - x)^(2)) Sum[EulerPhi[n]*Divide[(x)^(n),1 - (x)^(n)], {n, 1, Infinity}, GenerateConditions->None] == Divide[x,(1 - x)^(2)] Failure Successful Successful [Tested: 1] Successful [Tested: 1]
27.7.E5 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \sum_{n=1}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}} = \sum_{n=1}^{\infty}\sumdivisors{\alpha}@{n}x^{n}} sum((n)^(alpha)*((x)^(n))/(1 - (x)^(n)), n = 1..infinity) = sum(add(divisors(alpha))*(x)^(n), n = 1..infinity) Error Failure Missing Macro Error
Failed [3 / 3]
3/3]: [[2.671514971 <- {alpha = 3/2, x = 1/2}
1.507450946 <- {alpha = 1/2, x = 1/2}
 -
27.9.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Legendresym{-1}{p} = (-1)^{(p-1)/2}} LegendreSymbol(- 1, p) = (- 1)^((p - 1)/ 2) Error Failure Missing Macro Error Error -
27.9.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Legendresym{2}{p} = (-1)^{(p^{2}-1)/8}} LegendreSymbol(2, p) = (- 1)^(((p)^(2)- 1)/ 8) Error Failure Missing Macro Error Error -
27.9.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Legendresym{p}{q}\Legendresym{q}{p} = (-1)^{(p-1)(q-1)/4}} LegendreSymbol(p, q)*LegendreSymbol(q, p) = (- 1)^((p - 1)*(q - 1)/ 4) Error Failure Missing Macro Error Error -
27.10.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle s_{k}(n) = \sum_{m=1}^{k}a_{k}(m)e^{2\cpi\iunit mn/k}} s[k]*(n) = sum(a[k]*(m)* exp(2*Pi*I*m*n/ k), m = 1..k) Subscript[s, k]*(n) == Sum[Subscript[a, k]*(m)* Exp[2*Pi*I*m*n/ k], {m, 1, k}, GenerateConditions->None] Failure Failure
Failed [297 / 300]
297/300]: [[2971422279.-5146654356.*I <- {a[k] = 1/2*3^(1/2)+1/2*I, s[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}
-1114283352.+1929995386.*I <- {a[k] = 1/2*3^(1/2)+1/2*I, s[k] = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}
Failed [297 / 300]
{Indeterminate <- {Rule[k, 1], Rule[n, 1], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Indeterminate <- {Rule[k, 1], Rule[n, 2], Rule[Subscript[a, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[Subscript[s, k], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
27.12.E1 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \lim_{n\to\infty}\frac{p_{n}}{n\ln@@{n}} = 1} limit((p[n])/(n*ln(n)), n = infinity) = 1 Limit[Divide[Subscript[p, n],n*Log[n]], n -> Infinity, GenerateConditions->None] == 1 Failure Failure Skip - No test values generated
Failed [10 / 10]
{-1.0 <- {Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
-1.0 <- {Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}
27.12.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle p_{n} > n\ln@@{n}} p[n] > n*ln(n) Subscript[p, n] > n*Log[n] Failure Failure
Failed [6 / 10]
6/10]: [[3.295836867 < -1.500000000 <- {p[n] = -3/2, n = 3}
3.295836867 < 1.500000000 <- {p[n] = 3/2, n = 3}
Failed [25 / 30]
{Greater[Complex[0.8660254037844387, 0.49999999999999994], 0.0] <- {Rule[n, 1], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
Greater[Complex[0.8660254037844387, 0.49999999999999994], 1.3862943611198906] <- {Rule[n, 2], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}
27.13.E4 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \AThetaFunction@{x} = 1+2\sum_{m=1}^{\infty}x^{m^{2}}} 1+2*(sum((x)^(m^2), m = 1 .. infinity)) = 1 + 2*sum((x)^((m)^(2)), m = 1..infinity) Error Successful Missing Macro Error - -
27.13.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle (\AThetaFunction@{x})^{2} = 1+4\sum_{n=1}^{\infty}\left(\delta_{1}(n)-\delta_{3}(n)\right)x^{n}} (1+2*(sum((x)^(m^2), m = 1 .. infinity)))^(2) = 1 + 4*sum((delta[1]*(n)- delta[3]*(n))* (x)^(n), n = 1..infinity) Error Failure Missing Macro Error
Failed [300 / 300]
300/300]: [[3.532372013 <- {delta = 1/2*3^(1/2)+1/2*I, x = 1/2, delta[1] = 1/2*3^(1/2)+1/2*I, delta[3] = 1/2*3^(1/2)+1/2*I}
-7.395831219+2.928203232*I <- {delta = 1/2*3^(1/2)+1/2*I, x = 1/2, delta[1] = 1/2*3^(1/2)+1/2*I, delta[3] = -1/2+1/2*I*3^(1/2)}
 -
27.14.E2 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \EulerPhi@{x} = \prod_{m=1}^{\infty}(1-x^{m})} product(1-(x)^k, k = 1 .. infinity) = product(1 - (x)^(m), m = 1..infinity) QPochhammer[x,x] == Product[1 - (x)^(m), {m, 1, Infinity}, GenerateConditions->None] Failure Successful Successful [Tested: 1] Successful [Tested: 1]
27.14.E3 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \frac{1}{\EulerPhi@{x}} = \sum_{n=0}^{\infty}\npartitions[]@{n}x^{n}} (1)/(product(1-(x)^k, k = 1 .. infinity)) = sum(nops(partition(n))*(x)^(n), n = 0..infinity) Error Failure Missing Macro Error Error -
27.14.E6 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \npartitions[]@{n} = \sum_{k=1}^{\infty}(-1)^{k+1}\left(\npartitions[]@{n-\omega(k)}+\npartitions[]@{n-\omega(-k)}\right)} nops(partition(n)) = sum((- 1)^(k + 1)*(nops(partition(n - omega*(k)))+ nops(partition(n - omega*(- k)))), k = 1..infinity) Error Error Missing Macro Error - -
27.14.E7 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle n\npartitions[]@{n} = \sum_{k=1}^{n}\sumdivisors{1}@{k}\npartitions[]@{n-k}} n*nops(partition(n)) = sum(add(divisors(1))*nops(partition(n - k)), k = 1..n) Error Error Missing Macro Error - -
27.14.E9 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \npartitions[]@{n} = \frac{1}{\cpi\sqrt{2}}\sum_{k=1}^{\infty}\sqrt{k}A_{k}(n)\*\left[\deriv{}{t}\frac{\sinh@{\ifrac{K\sqrt{t}}{k}}}{\sqrt{t}}\right]_{t=n-(1/24)}} nops(partition(n)) = (1)/(Pi*sqrt(2))*sum(sqrt(k)*A[k]*(n)*[diff((sinh((K*sqrt(t))/(k)))/(sqrt(t)), t)][t = n -(1/ 24)], k = 1..infinity) Error Error Missing Macro Error - -
27.14.E12 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Dedekindeta@{\tau} = e^{\cpi\iunit\tau/12}\prod_{n=1}^{\infty}(1-e^{2\cpi\iunit n\tau})} Error DedekindEta[\[Tau]] == Exp[Pi*I*\[Tau]/ 12]*Product[1 - Exp[2*Pi*I*n*\[Tau]], {n, 1, Infinity}, GenerateConditions->None] Missing Macro Error Failure - Successful [Tested: 1]
27.14.E13 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Dedekindeta@{\tau} = e^{\cpi\iunit\tau/12}\EulerPhi@{e^{2\cpi\iunit\tau}}} Error DedekindEta[\[Tau]] == Exp[Pi*I*\[Tau]/ 12]*QPochhammer[Exp[2*Pi*I*\[Tau]],Exp[2*Pi*I*\[Tau]]] Missing Macro Error Failure - Successful [Tested: 1]
27.14.E14 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle \Dedekindeta@{\frac{a\tau+b}{c\tau+d}} = \varepsilon(-\iunit(c\tau+d))^{\frac{1}{2}}\Dedekindeta@{\tau}} Error DedekindEta[Divide[a*\[Tau]+ b,c*\[Tau]+ d]] == \[CurlyEpsilon]*(- I*(c*\[Tau]+ d))^(Divide[1,2])* DedekindEta[\[Tau]] Missing Macro Error Failure -
Failed [135 / 300]
{Complex[0.13319594449577687, -0.32363546143707655] <- {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[d, -2], Rule[ε, 1], Rule[τ, Complex[0, 1]]}
Complex[-0.41002146111087723, -1.4100702726503846] <- {Rule[a, -1.5], Rule[b, -1.5], Rule[c, -1.5], Rule[d, -2], Rule[ε, 2], Rule[τ, Complex[0, 1]]}
27.14.E15 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle 5\frac{(\EulerPhi@{x^{5}})^{5}}{(\EulerPhi@{x})^{6}} = \sum_{n=0}^{\infty}\npartitions[]@{5n+4}x^{n}} 5*((product(1-((x)^(5))^k, k = 1 .. infinity))^(5))/((product(1-(x)^k, k = 1 .. infinity))^(6)) = sum(nops(partition(5*n + 4))*(x)^(n), n = 0..infinity) Error Failure Missing Macro Error Error -
27.14.E18 Failed to parse (LaTeXML (experimental; uses MathML): Invalid response ("") from server "http://latexml:8080/convert/":): {\displaystyle x\prod_{n=1}^{\infty}(1-x^{n})^{24} = \sum_{n=1}^{\infty}\Ramanujantau@{n}x^{n}} Error x*Product[(1 - (x)^(n))^(24), {n, 1, Infinity}, GenerateConditions->None] == Sum[RamanujanTau[n]*(x)^(n), {n, 1, Infinity}, GenerateConditions->None] Missing Macro Error Successful - Successful [Tested: 1]
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