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 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 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 Error Error - -
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}] Successful Successful - -
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}]= Product[(1 - (p)^(- s))^(- 1), {p, - Infinity, Infinity}] Failure Failure Skip Error
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 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}} \|add(divisors(alpha))*(n)^(- s), n = 1..infinity)= Zeta(s)*Zeta(s - alpha) Error Error Error - -
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)= - subs( temp=s, diff( Zeta(temp), temp$(1) ) ) Sum[(Log[n])* (n)^(- s), {n, 2, Infinity}]= - (D[Zeta[temp], {temp, 1}]/.temp-> s) Successful Successful - -
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}} \|add(divisors(alpha))*(x)^(n), n = 1..infinity) Error Error Error - -
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}} legendre(- 1, p)=(- 1)^((p - 1)/ 2) Error 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}} legendre(2, p)=(- 1)^(((p)^(2)- 1)/ 8) Error 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}} legendre(p, q)*legendre(q, p)=(- 1)^((p - 1)*(q - 1)/ 4) Error 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}] Failure Failure Skip Successful
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]= 1 Successful Failure -
Fail
Complex[0.41421356237309515, 1.4142135623730951] <- {Rule[Limit[Times[Power[n, -1], Power[Log[n], -1], Subscript[p, n]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.41421356237309515, -1.4142135623730951] <- {Rule[Limit[Times[Power[n, -1], Power[Log[n], -1], Subscript[p, n]], Rule[n, DirectedInfinity[1]]], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.414213562373095, -1.4142135623730951] <- {Rule[Limit[Times[Power[n, -1], Power[Log[n], -1], Subscript[p, n]], Rule[n, DirectedInfinity[1]]], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.414213562373095, 1.4142135623730951] <- {Rule[Limit[Times[Power[n, -1], Power[Log[n], -1], Subscript[p, n]], Rule[n, DirectedInfinity[1]]], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
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 Successful Successful
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 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 Error Skip -
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) Error Successful 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}]= Sum[RamanujanTau[n]*(x)^(n), {n, 1, Infinity}] Error Successful - -
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