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 n = ∏ r = 1 ν ⁡ ( n ) p r a r 𝑛 superscript subscript product 𝑟 1 number-of-primes-dividing-nu 𝑛 subscript superscript 𝑝 subscript 𝑎 𝑟 𝑟 {\displaystyle{\displaystyle n=\prod_{r=1}^{\nu\left(n\right)}p^{a_{r}}_{r}}} n product(p(p[r])^(a[r]), r = 1..ifactor(n)) Error Error Translation Error - -
27.2.E7 ϕ ⁡ ( n ) = ϕ 0 ⁡ ( n ) Euler-totient-phi 𝑛 Euler-totient-phi-n 0 𝑛 {\displaystyle{\displaystyle\phi\left(n\right)=\phi_{0}\left(n\right)}} 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 d ⁡ ( n ) = ∑ d | n 1 divisor-function-D 𝑛 subscript divides 𝑑 𝑛 1 {\displaystyle{\displaystyle d\left(n\right)=\sum_{d\mathbin{|}n}1}} numelems(Divisors(n)) = sum(1, d**n in - infinity) Error Translation Error Missing Macro Error - -
27.2.E10 σ α ⁡ ( n ) = ∑ d | n d α divisor-sigma 𝛼 𝑛 subscript divides 𝑑 𝑛 superscript 𝑑 𝛼 {\displaystyle{\displaystyle\sigma_{\alpha}\left(n\right)=\sum_{d\mathbin{|}n}% d^{\alpha}}} add(divisors(alpha)) = sum((d)^(alpha), d**n in - infinity) Error Translation Error Missing Macro Error - -
27.3.E3 ϕ ⁡ ( n ) = n ⁢ ∏ p | n ( 1 - p - 1 ) Euler-totient-phi 𝑛 𝑛 subscript product divides 𝑝 𝑛 1 superscript 𝑝 1 {\displaystyle{\displaystyle\phi\left(n\right)=n\prod_{p\mathbin{|}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 d ⁡ ( n ) = ∏ r = 1 ν ⁡ ( n ) ( 1 + a r ) divisor-function-D 𝑛 superscript subscript product 𝑟 1 number-of-primes-dividing-nu 𝑛 1 subscript 𝑎 𝑟 {\displaystyle{\displaystyle d\left(n\right)=\prod_{r=1}^{\nu\left(n\right)}(1% +a_{r})}} numelems(Divisors(n)) = product(1 + a[r], r = 1..ifactor(n)) Error Error Missing Macro Error - -
27.3.E6 σ α ⁡ ( n ) = ∏ r = 1 ν ⁡ ( n ) p r α ⁢ ( 1 + a r ) - 1 p r α - 1 divisor-sigma 𝛼 𝑛 superscript subscript product 𝑟 1 number-of-primes-dividing-nu 𝑛 subscript superscript 𝑝 𝛼 1 subscript 𝑎 𝑟 𝑟 1 subscript superscript 𝑝 𝛼 𝑟 1 {\displaystyle{\displaystyle\sigma_{\alpha}\left(n\right)=\prod_{r=1}^{\nu% \left(n\right)}\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 ϕ ⁡ ( m ) ⁢ ϕ ⁡ ( n ) = ϕ ⁡ ( m ⁢ n ) ⁢ ϕ ⁡ ( ( m , n ) ) / ( m , n ) Euler-totient-phi 𝑚 Euler-totient-phi 𝑛 Euler-totient-phi 𝑚 𝑛 Euler-totient-phi 𝑚 𝑛 𝑚 𝑛 {\displaystyle{\displaystyle\phi\left(m\right)\phi\left(n\right)=\phi\left(mn% \right)\phi\left(\left(m,n\right)\right)/\left(m,n\right)}} 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 ζ ⁡ ( s ) = ∑ n = 1 ∞ n - s Riemann-zeta 𝑠 superscript subscript 𝑛 1 superscript 𝑛 𝑠 {\displaystyle{\displaystyle\zeta\left(s\right)=\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 ∑ n = 1 ∞ n - s = ∏ p ( 1 - p - s ) - 1 superscript subscript 𝑛 1 superscript 𝑛 𝑠 subscript product 𝑝 superscript 1 superscript 𝑝 𝑠 1 {\displaystyle{\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 ∑ n = 1 ∞ ϕ ⁡ ( n ) ⁢ n - s = ζ ⁡ ( s - 1 ) ζ ⁡ ( s ) superscript subscript 𝑛 1 Euler-totient-phi 𝑛 superscript 𝑛 𝑠 Riemann-zeta 𝑠 1 Riemann-zeta 𝑠 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}\phi\left(n\right)n^{-s}=\frac{% \zeta\left(s-1\right)}{\zeta\left(s\right)}}} 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 ∑ n = 1 ∞ 2 ν ⁡ ( n ) ⁢ n - s = ( ζ ⁡ ( s ) ) 2 ζ ⁡ ( 2 ⁢ s ) superscript subscript 𝑛 1 superscript 2 number-of-primes-dividing-nu 𝑛 superscript 𝑛 𝑠 superscript Riemann-zeta 𝑠 2 Riemann-zeta 2 𝑠 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}2^{\nu\left(n\right)}n^{-s}=% \frac{(\zeta\left(s\right))^{2}}{\zeta\left(2s\right)}}} sum((2)^(ifactor(n))* (n)^(- s), n = 1..infinity) = ((Zeta(s))^(2))/(Zeta(2*s)) Error Error Missing Macro Error - -
27.4.E11 ∑ n = 1 ∞ σ α ⁡ ( n ) ⁢ n - s = ζ ⁡ ( s ) ⁢ ζ ⁡ ( s - α ) superscript subscript 𝑛 1 divisor-sigma 𝛼 𝑛 superscript 𝑛 𝑠 Riemann-zeta 𝑠 Riemann-zeta 𝑠 𝛼 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)n^% {-s}=\zeta\left(s\right)\zeta\left(s-\alpha\right)}} 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 ∑ n = 2 ∞ ( ln ⁡ n ) ⁢ n - s = - ζ ′ ⁡ ( s ) superscript subscript 𝑛 2 𝑛 superscript 𝑛 𝑠 diffop Riemann-zeta 1 𝑠 {\displaystyle{\displaystyle\sum_{n=2}^{\infty}(\ln n)n^{-s}=-\zeta'\left(s% \right)}} 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 ∑ n = 1 ∞ ϕ ⁡ ( n ) ⁢ x n 1 - x n = x ( 1 - x ) 2 superscript subscript 𝑛 1 Euler-totient-phi 𝑛 superscript 𝑥 𝑛 1 superscript 𝑥 𝑛 𝑥 superscript 1 𝑥 2 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}\phi\left(n\right)\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 ∑ n = 1 ∞ n α ⁢ x n 1 - x n = ∑ n = 1 ∞ σ α ⁡ ( n ) ⁢ x n superscript subscript 𝑛 1 superscript 𝑛 𝛼 superscript 𝑥 𝑛 1 superscript 𝑥 𝑛 superscript subscript 𝑛 1 divisor-sigma 𝛼 𝑛 superscript 𝑥 𝑛 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}}% =\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)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 ( - 1 | p ) = ( - 1 ) ( p - 1 ) / 2 Legendre-symbol 1 𝑝 superscript 1 𝑝 1 2 {\displaystyle{\displaystyle(-1|p)=(-1)^{(p-1)/2}}} LegendreSymbol(- 1, p) = (- 1)^((p - 1)/ 2) Error Failure Missing Macro Error Error -
27.9.E2 ( 2 | p ) = ( - 1 ) ( p 2 - 1 ) / 8 Legendre-symbol 2 𝑝 superscript 1 superscript 𝑝 2 1 8 {\displaystyle{\displaystyle(2|p)=(-1)^{(p^{2}-1)/8}}} LegendreSymbol(2, p) = (- 1)^(((p)^(2)- 1)/ 8) Error Failure Missing Macro Error Error -
27.9.E3 ( p | q ) ⁢ ( q | p ) = ( - 1 ) ( p - 1 ) ⁢ ( q - 1 ) / 4 Legendre-symbol 𝑝 𝑞 Legendre-symbol 𝑞 𝑝 superscript 1 𝑝 1 𝑞 1 4 {\displaystyle{\displaystyle(p|q)(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 s k ⁢ ( n ) = ∑ m = 1 k a k ⁢ ( m ) ⁢ e 2 ⁢ π ⁢ i ⁢ m ⁢ n / k subscript 𝑠 𝑘 𝑛 superscript subscript 𝑚 1 𝑘 subscript 𝑎 𝑘 𝑚 superscript 𝑒 2 imaginary-unit 𝑚 𝑛 𝑘 {\displaystyle{\displaystyle s_{k}(n)=\sum_{m=1}^{k}a_{k}(m)e^{2\pi\mathrm{i}% 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 lim n → ∞ ⁡ p n n ⁢ ln ⁡ n = 1 subscript → 𝑛 subscript 𝑝 𝑛 𝑛 𝑛 1 {\displaystyle{\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 p n > n ⁢ ln ⁡ n subscript 𝑝 𝑛 𝑛 𝑛 {\displaystyle{\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 ϑ ⁡ ( x ) = 1 + 2 ⁢ ∑ m = 1 ∞ x m 2 a-theta-function 𝑥 1 2 superscript subscript 𝑚 1 superscript 𝑥 superscript 𝑚 2 {\displaystyle{\displaystyle\vartheta\left(x\right)=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 ( ϑ ⁡ ( x ) ) 2 = 1 + 4 ⁢ ∑ n = 1 ∞ ( δ 1 ⁢ ( n ) - δ 3 ⁢ ( n ) ) ⁢ x n superscript a-theta-function 𝑥 2 1 4 superscript subscript 𝑛 1 subscript 𝛿 1 𝑛 subscript 𝛿 3 𝑛 superscript 𝑥 𝑛 {\displaystyle{\displaystyle(\vartheta\left(x\right))^{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 f ⁡ ( x ) = ∏ m = 1 ∞ ( 1 - x m ) Euler-phi 𝑥 superscript subscript product 𝑚 1 1 superscript 𝑥 𝑚 {\displaystyle{\displaystyle\mathit{f}\left(x\right)=\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 1 f ⁡ ( x ) = ∑ n = 0 ∞ p ⁡ ( n ) ⁢ x n 1 Euler-phi 𝑥 superscript subscript 𝑛 0 partition-function 𝑛 superscript 𝑥 𝑛 {\displaystyle{\displaystyle\frac{1}{\mathit{f}\left(x\right)}=\sum_{n=0}^{% \infty}p\left(n\right)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 p ⁡ ( n ) = ∑ k = 1 ∞ ( - 1 ) k + 1 ⁢ ( p ⁡ ( n - ω ⁢ ( k ) ) + p ⁡ ( n - ω ⁢ ( - k ) ) ) partition-function 𝑛 superscript subscript 𝑘 1 superscript 1 𝑘 1 partition-function 𝑛 𝜔 𝑘 partition-function 𝑛 𝜔 𝑘 {\displaystyle{\displaystyle p\left(n\right)=\sum_{k=1}^{\infty}(-1)^{k+1}% \left(p\left(n-\omega(k)\right)+p\left(n-\omega(-k)\right)\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 n ⁢ p ⁡ ( n ) = ∑ k = 1 n σ 1 ⁡ ( k ) ⁢ p ⁡ ( n - k ) 𝑛 partition-function 𝑛 superscript subscript 𝑘 1 𝑛 divisor-sigma 1 𝑘 partition-function 𝑛 𝑘 {\displaystyle{\displaystyle np\left(n\right)=\sum_{k=1}^{n}\sigma_{1}\left(k% \right)p\left(n-k\right)}} n*nops(partition(n)) = sum(add(divisors(1))*nops(partition(n - k)), k = 1..n) Error Error Missing Macro Error - -
27.14.E9 p ⁡ ( n ) = 1 π ⁢ 2 ⁢ ∑ k = 1 ∞ k ⁢ A k ⁢ ( n ) ⁢ [ d d t ⁡ sinh ⁡ ( K ⁢ t / k ) t ] t = n - ( 1 / 24 ) partition-function 𝑛 1 2 superscript subscript 𝑘 1 𝑘 subscript 𝐴 𝑘 𝑛 subscript delimited-[] derivative 𝑡 𝐾 𝑡 𝑘 𝑡 𝑡 𝑛 1 24 {\displaystyle{\displaystyle p\left(n\right)=\frac{1}{\pi\sqrt{2}}\sum_{k=1}^{% \infty}\sqrt{k}A_{k}(n)\*\left[\frac{\mathrm{d}}{\mathrm{d}t}\frac{\sinh\left(% \ifrac{K\sqrt{t}}{k}\right)}{\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 η ⁡ ( τ ) = e π ⁢ i ⁢ τ / 12 ⁢ ∏ n = 1 ∞ ( 1 - e 2 ⁢ π ⁢ i ⁢ n ⁢ τ ) Dedekind-modular-Eta 𝜏 superscript 𝑒 imaginary-unit 𝜏 12 superscript subscript product 𝑛 1 1 superscript 𝑒 2 imaginary-unit 𝑛 𝜏 {\displaystyle{\displaystyle\eta\left(\tau\right)=e^{\pi\mathrm{i}\tau/12}% \prod_{n=1}^{\infty}(1-e^{2\pi\mathrm{i}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 η ⁡ ( τ ) = e π ⁢ i ⁢ τ / 12 ⁢ f ⁡ ( e 2 ⁢ π ⁢ i ⁢ τ ) Dedekind-modular-Eta 𝜏 superscript 𝑒 imaginary-unit 𝜏 12 Euler-phi superscript 𝑒 2 imaginary-unit 𝜏 {\displaystyle{\displaystyle\eta\left(\tau\right)=e^{\pi\mathrm{i}\tau/12}% \mathit{f}\left(e^{2\pi\mathrm{i}\tau}\right)}} 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 η ⁡ ( a ⁢ τ + b c ⁢ τ + d ) = ε ⁢ ( - i ⁢ ( c ⁢ τ + d ) ) 1 2 ⁢ η ⁡ ( τ ) Dedekind-modular-Eta 𝑎 𝜏 𝑏 𝑐 𝜏 𝑑 𝜀 superscript imaginary-unit 𝑐 𝜏 𝑑 1 2 Dedekind-modular-Eta 𝜏 {\displaystyle{\displaystyle\eta\left(\frac{a\tau+b}{c\tau+d}\right)=% \varepsilon(-\mathrm{i}(c\tau+d))^{\frac{1}{2}}\eta\left(\tau\right)}} 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 5 ⁢ ( f ⁡ ( x 5 ) ) 5 ( f ⁡ ( x ) ) 6 = ∑ n = 0 ∞ p ⁡ ( 5 ⁢ n + 4 ) ⁢ x n 5 superscript Euler-phi superscript 𝑥 5 5 superscript Euler-phi 𝑥 6 superscript subscript 𝑛 0 partition-function 5 𝑛 4 superscript 𝑥 𝑛 {\displaystyle{\displaystyle 5\frac{(\mathit{f}\left(x^{5}\right))^{5}}{(% \mathit{f}\left(x\right))^{6}}=\sum_{n=0}^{\infty}p\left(5n+4\right)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 x ⁢ ∏ n = 1 ∞ ( 1 - x n ) 24 = ∑ n = 1 ∞ τ ⁡ ( n ) ⁢ x n 𝑥 superscript subscript product 𝑛 1 superscript 1 superscript 𝑥 𝑛 24 superscript subscript 𝑛 1 Ramanujan-tau 𝑛 superscript 𝑥 𝑛 {\displaystyle{\displaystyle x\prod_{n=1}^{\infty}(1-x^{n})^{24}=\sum_{n=1}^{% \infty}\tau\left(n\right)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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