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| | ; Notation : [[27.1|27.1 Special Notation]]<br> |
| ! DLMF !! Formula !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
| | ; Multiplicative Number Theory : [[27.2|27.2 Functions]]<br>[[27.3|27.3 Multiplicative Properties]]<br>[[27.4|27.4 Euler Products and Dirichlet Series]]<br>[[27.5|27.5 Inversion Formulas]]<br>[[27.6|27.6 Divisor Sums]]<br>[[27.7|27.7 Lambert Series as Generating Functions]]<br>[[27.8|27.8 Dirichlet Characters]]<br>[[27.9|27.9 Quadratic Characters]]<br>[[27.10|27.10 Periodic Number-Theoretic Functions]]<br>[[27.11|27.11 Asymptotic Formulas: Partial Sums]]<br>[[27.12|27.12 Asymptotic Formulas: Primes]]<br> |
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| | ; Additive Number Theory : [[27.13|27.13 Functions]]<br>[[27.14|27.14 Unrestricted Partitions]]<br> |
| | [https://dlmf.nist.gov/27.2.E1 27.2.E1] || [[Item:Q7988|<math>n = \prod_{r=1}^{\nprimesdiv@{n}}p^{a_{r}}_{r}</math>]] || <code>n product(p(p[r])^(a[r]), r = 1..ifactor(n))</code> || <code>Error</code> || Error || Translation Error || - || -
| | ; Applications : [[27.15|27.15 Chinese Remainder Theorem]]<br>[[27.16|27.16 Cryptography]]<br>[[27.17|27.17 Other Applications]]<br> |
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| | ; Computation : [[27.18|27.18 Methods of Computation: Primes]]<br>[[27.19|27.19 Methods of Computation: Factorization]]<br>[[27.20|27.20 Methods of Computation: Other Number-Theoretic Functions]]<br>[[27.21|27.21 Tables]]<br>[[27.22|27.22 Software]]<br> |
| | [https://dlmf.nist.gov/27.2.E7 27.2.E7] || [[Item:Q7994|<math>\Eulertotientphi[]@{n} = \Eulertotientphi[0]@{n}</math>]] || <code>Error</code> || <code>EulerPhi[n] == Sum[If[CoprimeQ[n, m], m^(0), 0], {m, 1, n}]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{1.0 <- {Rule[n, 1]}</code><br><code>1.0 <- {Rule[n, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/27.2.E9 27.2.E9] || [[Item:Q7996|<math>\ndivisors[]@{n} = \sum_{d\divides n}1</math>]] || <code>numelems(Divisors(n)) = sum(1, d**n in - infinity)</code> || <code>Error</code> || Translation Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/27.2.E10 27.2.E10] || [[Item:Q7997|<math>\sumdivisors{\alpha}@{n} = \sum_{d\divides n}d^{\alpha}</math>]] || <code>add(divisors(alpha)) = sum((d)^(alpha), d**n in - infinity)</code> || <code>Error</code> || Translation Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/27.3.E3 27.3.E3] || [[Item:Q8004|<math>\Eulertotientphi[]@{n} = n\prod_{p\divides n}(1-p^{-1})</math>]] || <code>phi(n) = n*product(1 - (p)^(- 1), p**n in - infinity)</code> || <code>EulerPhi[n] == n*Product[1 - (p)^(- 1), {p**n, - Infinity}, GenerateConditions->None]</code> || Translation Error || Translation Error || - || -
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| | [https://dlmf.nist.gov/27.3.E5 27.3.E5] || [[Item:Q8006|<math>\ndivisors[]@{n} = \prod_{r=1}^{\nprimesdiv@{n}}(1+a_{r})</math>]] || <code>numelems(Divisors(n)) = product(1 + a[r], r = 1..ifactor(n))</code> || <code>Error</code> || Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/27.3.E6 27.3.E6] || [[Item:Q8007|<math>\sumdivisors{\alpha}@{n} = \prod_{r=1}^{\nprimesdiv@{n}}\frac{p^{\alpha(1+a_{r})}_{r}-1}{p^{\alpha}_{r}-1}</math>]] || <code>product((p(p[r])^(alpha*(1 + a[r]))- 1)/(p(p[r])^(alpha)- 1), r = 1..ifactor(n))</code> || <code>Error</code> || Failure || Missing Macro Error || Error || - | |
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| | [https://dlmf.nist.gov/27.3.E8 27.3.E8] || [[Item:Q8009|<math>\Eulertotientphi[]@{m}\Eulertotientphi[]@{n} = \Eulertotientphi[]@{mn}\Eulertotientphi[]@{\pgcd{m,n}}/\pgcd{m,n}</math>]] || <code>phi(m)*phi(n) = phi(m*n)*phi(gcd(m , n))/ gcd(m , n)</code> || <code>EulerPhi[m]*EulerPhi[n] == EulerPhi[m*n]*EulerPhi[GCD[m , n]]/ GCD[m , n]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 9]<div class="mw-collapsible-content"><code>2/9]: [[-1. <- {m = 2, n = 2}</code><br><code>-2. <- {m = 3, n = 3}</code><br></div></div> || Successful [Tested: 9]
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| | [https://dlmf.nist.gov/27.4.E3 27.4.E3] || [[Item:Q8014|<math>\Riemannzeta@{s} = \sum_{n=1}^{\infty}n^{-s}</math>]] || <code>Zeta(s) = sum((n)^(- s), n = 1..infinity)</code> || <code>Zeta[s] == Sum[(n)^(- s), {n, 1, Infinity}, GenerateConditions->None]</code> || Successful || Successful || Skip - symbolical successful subtest || Successful [Tested: 2]
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| | [https://dlmf.nist.gov/27.4.E3 27.4.E3] || [[Item:Q8014|<math>\sum_{n=1}^{\infty}n^{-s} = \prod_{p}(1-p^{-s})^{-1}</math>]] || <code>sum((n)^(- s), n = 1..infinity) = product((1 - (p)^(- s))^(- 1), p = - infinity..infinity)</code> || <code>Sum[(n)^(- s), {n, 1, Infinity}, GenerateConditions->None] == Product[(1 - (p)^(- s))^(- 1), {p, - Infinity, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 2]<div class="mw-collapsible-content"><code>{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]}</code><br><code>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]}</code><br></div></div>
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| | [https://dlmf.nist.gov/27.4.E6 27.4.E6] || [[Item:Q8017|<math>\sum_{n=1}^{\infty}\Eulertotientphi[]@{n}n^{-s} = \frac{\Riemannzeta@{s-1}}{\Riemannzeta@{s}}</math>]] || <code>sum(phi(n)*(n)^(- s), n = 1..infinity) = (Zeta(s - 1))/(Zeta(s))</code> || <code>Sum[EulerPhi[n]*(n)^(- s), {n, 1, Infinity}, GenerateConditions->None] == Divide[Zeta[s - 1],Zeta[s]]</code> || Failure || Successful || Error || Successful [Tested: 0]
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| | [https://dlmf.nist.gov/27.4.E9 27.4.E9] || [[Item:Q8020|<math>\sum_{n=1}^{\infty}2^{\nprimesdiv@{n}}n^{-s} = \frac{(\Riemannzeta@{s})^{2}}{\Riemannzeta@{2s}}</math>]] || <code>sum((2)^(ifactor(n))* (n)^(- s), n = 1..infinity) = ((Zeta(s))^(2))/(Zeta(2*s))</code> || <code>Error</code> || Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/27.4.E11 27.4.E11] || [[Item:Q8022|<math>\sum_{n=1}^{\infty}\sumdivisors{\alpha}@{n}n^{-s} = \Riemannzeta@{s}\Riemannzeta@{s-\alpha}</math>]] || <code>sum(add(divisors(alpha))*(n)^(- s), n = 1..infinity) = Zeta(s)*Zeta(s - alpha)</code> || <code>Error</code> || Failure || Missing Macro Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [18 / 18]<div class="mw-collapsible-content"><code>18/18]: [[Float(infinity) <- {alpha = 3/2, s = -3/2}</code><br><code>5.224750698 <- {alpha = 3/2, s = 3/2}</code><br></div></div> || -
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| | [https://dlmf.nist.gov/27.4.E13 27.4.E13] || [[Item:Q8024|<math>\sum_{n=2}^{\infty}(\ln@@{n})n^{-s} = -\Riemannzeta'@{s}</math>]] || <code>sum((ln(n))* (n)^(- s), n = 2..infinity) = - diff( Zeta(s), s$(1) )</code> || <code>Sum[(Log[n])* (n)^(- s), {n, 2, Infinity}, GenerateConditions->None] == - D[Zeta[s], {s, 1}]</code> || Successful || Successful || - || Successful [Tested: 2]
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| | [https://dlmf.nist.gov/27.7.E4 27.7.E4] || [[Item:Q8044|<math>\sum_{n=1}^{\infty}\Eulertotientphi[]@{n}\frac{x^{n}}{1-x^{n}} = \frac{x}{(1-x)^{2}}</math>]] || <code>sum(phi(n)*((x)^(n))/(1 - (x)^(n)), n = 1..infinity) = (x)/((1 - x)^(2))</code> || <code>Sum[EulerPhi[n]*Divide[(x)^(n),1 - (x)^(n)], {n, 1, Infinity}, GenerateConditions->None] == Divide[x,(1 - x)^(2)]</code> || Failure || Successful || Successful [Tested: 1] || Successful [Tested: 1]
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| | [https://dlmf.nist.gov/27.7.E5 27.7.E5] || [[Item:Q8045|<math>\sum_{n=1}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}} = \sum_{n=1}^{\infty}\sumdivisors{\alpha}@{n}x^{n}</math>]] || <code>sum((n)^(alpha)*((x)^(n))/(1 - (x)^(n)), n = 1..infinity) = sum(add(divisors(alpha))*(x)^(n), n = 1..infinity)</code> || <code>Error</code> || Failure || Missing Macro Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>3/3]: [[2.671514971 <- {alpha = 3/2, x = 1/2}</code><br><code>1.507450946 <- {alpha = 1/2, x = 1/2}</code><br></div></div> || -
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| | [https://dlmf.nist.gov/27.9.E1 27.9.E1] || [[Item:Q8055|<math>\Legendresym{-1}{p} = (-1)^{(p-1)/2}</math>]] || <code>LegendreSymbol(- 1, p) = (- 1)^((p - 1)/ 2)</code> || <code>Error</code> || Failure || Missing Macro Error || Error || -
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| | [https://dlmf.nist.gov/27.9.E2 27.9.E2] || [[Item:Q8056|<math>\Legendresym{2}{p} = (-1)^{(p^{2}-1)/8}</math>]] || <code>LegendreSymbol(2, p) = (- 1)^(((p)^(2)- 1)/ 8)</code> || <code>Error</code> || Failure || Missing Macro Error || Error || -
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| | [https://dlmf.nist.gov/27.9.E3 27.9.E3] || [[Item:Q8057|<math>\Legendresym{p}{q}\Legendresym{q}{p} = (-1)^{(p-1)(q-1)/4}</math>]] || <code>LegendreSymbol(p, q)*LegendreSymbol(q, p) = (- 1)^((p - 1)*(q - 1)/ 4)</code> || <code>Error</code> || Failure || Missing Macro Error || Error || -
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| | [https://dlmf.nist.gov/27.10.E7 27.10.E7] || [[Item:Q8064|<math>s_{k}(n) = \sum_{m=1}^{k}a_{k}(m)e^{2\cpi\iunit mn/k}</math>]] || <code>s[k]*(n) = sum(a[k]*(m)* exp(2*Pi*I*m*n/ k), m = 1..k)</code> || <code>Subscript[s, k]*(n) == Sum[Subscript[a, k]*(m)* Exp[2*Pi*I*m*n/ k], {m, 1, k}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [297 / 300]<div class="mw-collapsible-content"><code>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}</code><br><code>-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}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [297 / 300]<div class="mw-collapsible-content"><code>{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]]]}</code><br><code>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]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/27.12.E1 27.12.E1] || [[Item:Q8085|<math>\lim_{n\to\infty}\frac{p_{n}}{n\ln@@{n}} = 1</math>]] || <code>limit((p[n])/(n*ln(n)), n = infinity) = 1</code> || <code>Limit[Divide[Subscript[p, n],n*Log[n]], n -> Infinity, GenerateConditions->None] == 1</code> || Failure || Failure || Skip - No test values generated || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{-1.0 <- {Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>-1.0 <- {Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/27.12.E2 27.12.E2] || [[Item:Q8086|<math>p_{n} > n\ln@@{n}</math>]] || <code>p[n] > n*ln(n)</code> || <code>Subscript[p, n] > n*Log[n]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 10]<div class="mw-collapsible-content"><code>6/10]: [[3.295836867 < -1.500000000 <- {p[n] = -3/2, n = 3}</code><br><code>3.295836867 < 1.500000000 <- {p[n] = 3/2, n = 3}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [25 / 30]<div class="mw-collapsible-content"><code>{Greater[Complex[0.8660254037844387, 0.49999999999999994], 0.0] <- {Rule[n, 1], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Greater[Complex[0.8660254037844387, 0.49999999999999994], 1.3862943611198906] <- {Rule[n, 2], Rule[Subscript[p, n], Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/27.13.E4 27.13.E4] || [[Item:Q8096|<math>\AThetaFunction@{x} = 1+2\sum_{m=1}^{\infty}x^{m^{2}}</math>]] || <code>1+2*(sum((x)^(m^2), m = 1 .. infinity)) = 1 + 2*sum((x)^((m)^(2)), m = 1..infinity)</code> || <code>Error</code> || Successful || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/27.13.E6 27.13.E6] || [[Item:Q8098|<math>(\AThetaFunction@{x})^{2} = 1+4\sum_{n=1}^{\infty}\left(\delta_{1}(n)-\delta_{3}(n)\right)x^{n}</math>]] || <code>(1+2*(sum((x)^(m^2), m = 1 .. infinity)))^(2) = 1 + 4*sum((delta[1]*(n)- delta[3]*(n))* (x)^(n), n = 1..infinity)</code> || <code>Error</code> || Failure || Missing Macro Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>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}</code><br><code>-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)}</code><br></div></div> || -
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| | [https://dlmf.nist.gov/27.14.E2 27.14.E2] || [[Item:Q8101|<math>\EulerPhi@{x} = \prod_{m=1}^{\infty}(1-x^{m})</math>]] || <code>product(1-(x)^k, k = 1 .. infinity) = product(1 - (x)^(m), m = 1..infinity)</code> || <code>QPochhammer[x,x] == Product[1 - (x)^(m), {m, 1, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 1] || Successful [Tested: 1]
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| | [https://dlmf.nist.gov/27.14.E3 27.14.E3] || [[Item:Q8102|<math>\frac{1}{\EulerPhi@{x}} = \sum_{n=0}^{\infty}\npartitions[]@{n}x^{n}</math>]] || <code>(1)/(product(1-(x)^k, k = 1 .. infinity)) = sum(nops(partition(n))*(x)^(n), n = 0..infinity)</code> || <code>Error</code> || Failure || Missing Macro Error || Error || -
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| | [https://dlmf.nist.gov/27.14.E6 27.14.E6] || [[Item:Q8105|<math>\npartitions[]@{n} = \sum_{k=1}^{\infty}(-1)^{k+1}\left(\npartitions[]@{n-\omega(k)}+\npartitions[]@{n-\omega(-k)}\right)</math>]] || <code>nops(partition(n)) = sum((- 1)^(k + 1)*(nops(partition(n - omega*(k)))+ nops(partition(n - omega*(- k)))), k = 1..infinity)</code> || <code>Error</code> || Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/27.14.E7 27.14.E7] || [[Item:Q8106|<math>n\npartitions[]@{n} = \sum_{k=1}^{n}\sumdivisors{1}@{k}\npartitions[]@{n-k}</math>]] || <code>n*nops(partition(n)) = sum(add(divisors(1))*nops(partition(n - k)), k = 1..n)</code> || <code>Error</code> || Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/27.14.E9 27.14.E9] || [[Item:Q8108|<math>\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)}</math>]] || <code>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)</code> || <code>Error</code> || Error || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/27.14.E12 27.14.E12] || [[Item:Q8111|<math>\Dedekindeta@{\tau} = e^{\cpi\iunit\tau/12}\prod_{n=1}^{\infty}(1-e^{2\cpi\iunit n\tau})</math>]] || <code>Error</code> || <code>DedekindEta[\[Tau]] == Exp[Pi*I*\[Tau]/ 12]*Product[1 - Exp[2*Pi*I*n*\[Tau]], {n, 1, Infinity}, GenerateConditions->None]</code> || Missing Macro Error || Failure || - || Successful [Tested: 1] | |
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| | [https://dlmf.nist.gov/27.14.E13 27.14.E13] || [[Item:Q8112|<math>\Dedekindeta@{\tau} = e^{\cpi\iunit\tau/12}\EulerPhi@{e^{2\cpi\iunit\tau}}</math>]] || <code>Error</code> || <code>DedekindEta[\[Tau]] == Exp[Pi*I*\[Tau]/ 12]*QPochhammer[Exp[2*Pi*I*\[Tau]],Exp[2*Pi*I*\[Tau]]]</code> || Missing Macro Error || Failure || - || Successful [Tested: 1]
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| | [https://dlmf.nist.gov/27.14.E14 27.14.E14] || [[Item:Q8113|<math>\Dedekindeta@{\frac{a\tau+b}{c\tau+d}} = \varepsilon(-\iunit(c\tau+d))^{\frac{1}{2}}\Dedekindeta@{\tau}</math>]] || <code>Error</code> || <code>DedekindEta[Divide[a*\[Tau]+ b,c*\[Tau]+ d]] == \[CurlyEpsilon]*(- I*(c*\[Tau]+ d))^(Divide[1,2])* DedekindEta[\[Tau]]</code> || Missing Macro Error || Failure || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [135 / 300]<div class="mw-collapsible-content"><code>{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]]}</code><br><code>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]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/27.14.E15 27.14.E15] || [[Item:Q8114|<math>5\frac{(\EulerPhi@{x^{5}})^{5}}{(\EulerPhi@{x})^{6}} = \sum_{n=0}^{\infty}\npartitions[]@{5n+4}x^{n}</math>]] || <code>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)</code> || <code>Error</code> || Failure || Missing Macro Error || Error || -
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| | [https://dlmf.nist.gov/27.14.E18 27.14.E18] || [[Item:Q8117|<math>x\prod_{n=1}^{\infty}(1-x^{n})^{24} = \sum_{n=1}^{\infty}\Ramanujantau@{n}x^{n}</math>]] || <code>Error</code> || <code>x*Product[(1 - (x)^(n))^(24), {n, 1, Infinity}, GenerateConditions->None] == Sum[RamanujanTau[n]*(x)^(n), {n, 1, Infinity}, GenerateConditions->None]</code> || Missing Macro Error || Successful || - || Successful [Tested: 1]
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