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 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 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 Error Error - -
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}] Successful Successful - -
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}]= Product[(1 - (p)^(- s))^(- 1), {p, - Infinity, Infinity}] Failure Failure Skip Error
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 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)}} \|add(divisors(alpha))*(n)^(- s), n = 1..infinity)= Zeta(s)*Zeta(s - alpha) Error Error Error - -
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)= - 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 ∑ 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}}} \|add(divisors(alpha))*(x)^(n), n = 1..infinity) Error Error Error - -
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}}} legendre(- 1, p)=(- 1)^((p - 1)/ 2) Error 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}}} legendre(2, p)=(- 1)^(((p)^(2)- 1)/ 8) Error 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}}} legendre(p, q)*legendre(q, p)=(- 1)^((p - 1)*(q - 1)/ 4) Error 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}] Failure Failure Skip Successful
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]= 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 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 Successful Successful
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 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 Error Skip -
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) Error Successful 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}]= Sum[RamanujanTau[n]*(x)^(n), {n, 1, Infinity}] Error Successful - -
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