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| |- | | ; Notation : [[25.1|25.1 Special Notation]]<br> |
| ! DLMF !! Formula !! Maple !! Mathematica !! Symbolic<br>Maple !! Symbolic<br>Mathematica !! Numeric<br>Maple !! Numeric<br>Mathematica
| | ; Riemann Zeta Function : [[25.2|25.2 Definition and Expansions]]<br>[[25.3|25.3 Graphics]]<br>[[25.4|25.4 Reflection Formulas]]<br>[[25.5|25.5 Integral Representations]]<br>[[25.6|25.6 Integer Arguments]]<br>[[25.7|25.7 Integrals]]<br>[[25.8|25.8 Sums]]<br>[[25.9|25.9 Asymptotic Approximations]]<br>[[25.10|25.10 Zeros]]<br> |
| |-
| | ; Related Functions : [[25.11|25.11 Hurwitz Zeta Function]]<br>[[25.12|25.12 Polylogarithms]]<br>[[25.13|25.13 Periodic Zeta Function]]<br>[[25.14|25.14 Lerch’s Transcendent]]<br>[[25.15|25.15 Dirichlet <math>\DirichletL</math> -functions]]<br> |
| | [https://dlmf.nist.gov/25.2.E1 25.2.E1] || [[Item:Q7596|<math>\Riemannzeta@{s} = \sum_{n=1}^{\infty}\frac{1}{n^{s}}</math>]] || <code>Zeta(s) = sum((1)/((n)^(s)), n = 1..infinity)</code> || <code>Zeta[s] == Sum[Divide[1,(n)^(s)], {n, 1, Infinity}, GenerateConditions->None]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 6]<div class="mw-collapsible-content"><code>4/6]: [[Float(-infinity) <- {s = -3/2}</code><br><code>Float(-infinity) <- {s = -1/2}</code><br></div></div> || Successful [Tested: 6]
| | ; Applications : [[25.16|25.16 Mathematical Applications]]<br>[[25.17|25.17 Physical Applications]]<br> |
| |-
| | ; Computation : [[25.18|25.18 Methods of Computation]]<br>[[25.19|25.19 Tables]]<br>[[25.20|25.20 Approximations]]<br>[[25.21|25.21 Software]]<br> |
| | [https://dlmf.nist.gov/25.2.E2 25.2.E2] || [[Item:Q7597|<math>\Riemannzeta@{s} = \frac{1}{1-2^{-s}}\sum_{n=0}^{\infty}\frac{1}{(2n+1)^{s}}</math>]] || <code>Zeta(s) = (1)/(1 - (2)^(- s))*sum((1)/((2*n + 1)^(s)), n = 0..infinity)</code> || <code>Zeta[s] == Divide[1,1 - (2)^(- s)]*Sum[Divide[1,(2*n + 1)^(s)], {n, 0, Infinity}, GenerateConditions->None]</code> || Successful || Successful || - || Successful [Tested: 2]
| | </div> |
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| | [https://dlmf.nist.gov/25.2.E3 25.2.E3] || [[Item:Q7598|<math>\Riemannzeta@{s} = \frac{1}{1-2^{1-s}}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{s}}</math>]] || <code>Zeta(s) = (1)/(1 - (2)^(1 - s))*sum(((- 1)^(n - 1))/((n)^(s)), n = 1..infinity)</code> || <code>Zeta[s] == Divide[1,1 - (2)^(1 - s)]*Sum[Divide[(- 1)^(n - 1),(n)^(s)], {n, 1, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/25.2.E4 25.2.E4] || [[Item:Q7599|<math>\Riemannzeta@{s} = \frac{1}{s-1}+\sum_{n=0}^{\infty}\frac{(-1)^{n}}{n!}\StieltjesConstants{n}(s-1)^{n}</math>]] || <code>Zeta(s) = (1)/(s - 1)+ sum(((- 1)^(n))/(factorial(n))*gamma(n)*(s - 1)^(n), n = 0..infinity)</code> || <code>Error</code> || Failure || Missing Macro Error || Successful [Tested: 6] || -
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| | [https://dlmf.nist.gov/25.2.E5 25.2.E5] || [[Item:Q7600|<math>\StieltjesConstants{n} = \lim_{m\to\infty}\left(\sum_{k=1}^{m}\frac{(\ln@@{k})^{n}}{k}-\frac{(\ln@@{m})^{n+1}}{n+1}\right)</math>]] || <code>gamma(n) = limit(sum(((ln(k))^(n))/(k), k = 1..m)-((ln(m))^(n + 1))/(n + 1), m = infinity)</code> || <code>Error</code> || Successful || Missing Macro Error || - || -
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| | [https://dlmf.nist.gov/25.2.E6 25.2.E6] || [[Item:Q7601|<math>\Riemannzeta'@{s} = -\sum_{n=2}^{\infty}(\ln@@{n})n^{-s}</math>]] || <code>diff( Zeta(s), s$(1) ) = - sum((ln(n))* (n)^(- s), n = 2..infinity)</code> || <code>D[Zeta[s], {s, 1}] == - Sum[(Log[n])* (n)^(- s), {n, 2, Infinity}, GenerateConditions->None]</code> || Successful || Successful || - || Successful [Tested: 2]
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| | [https://dlmf.nist.gov/25.2.E7 25.2.E7] || [[Item:Q7602|<math>\Riemannzeta^{(k)}@{s} = (-1)^{k}\sum_{n=2}^{\infty}(\ln@@{n})^{k}n^{-s}</math>]] || <code>diff( Zeta(s), s$(k) ) = (- 1)^(k)* sum((ln(n))^(k)* (n)^(- s), n = 2..infinity)</code> || <code>D[Zeta[s], {s, k}] == (- 1)^(k)* Sum[(Log[n])^(k)* (n)^(- s), {n, 2, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
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| | [https://dlmf.nist.gov/25.2.E8 25.2.E8] || [[Item:Q7603|<math>\Riemannzeta@{s} = \sum_{k=1}^{N}\frac{1}{k^{s}}+\frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\floor{x}}{x^{s+1}}\diff{x}</math>]] || <code>Zeta(s) = sum((1)/((k)^(s)), k = 1..N)+((N)^(1 - s))/(s - 1)- s*int((x - floor(x))/((x)^(s + 1)), x = N..infinity)</code> || <code>Zeta[s] == Sum[Divide[1,(k)^(s)], {k, 1, N}, GenerateConditions->None]+Divide[(N)^(1 - s),s - 1]- s*Integrate[Divide[x - Floor[x],(x)^(s + 1)], {x, N, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>3/3]: [[.2180864797 <- {s = 3/2, N = 3}</code><br><code>Float(infinity) <- {s = 1/2, N = 3}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/25.2.E11 25.2.E11] || [[Item:Q7606|<math>\Riemannzeta@{s} = \prod_{p}(1-p^{-s})^{-1}</math>]] || <code>Zeta(s) = product((1 - (p)^(- s))^(- 1), p = - infinity..infinity)</code> || <code>Zeta[s] == 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/25.2.E12 25.2.E12] || [[Item:Q7607|<math>\Riemannzeta@{s} = \frac{(2\pi)^{s}e^{-s-(\EulerConstant s/2)}}{2(s-1)\EulerGamma@{\tfrac{1}{2}s+1}}\prod_{\rho}\left(1-\frac{s}{\rho}\right)e^{s/\rho}</math>]] || <code>Zeta(s) = ((2*Pi)^(s)* exp(- s -(gamma*s/ 2)))/(2*(s - 1)* GAMMA((1)/(2)*s + 1))*product((1 -(s)/(rho))* exp(s/ rho), rho = - infinity..infinity)</code> || <code>Zeta[s] == Divide[(2*Pi)^(s)* Exp[- s -(EulerGamma*s/ 2)],2*(s - 1)* Gamma[Divide[1,2]*s + 1]]*Product[(1 -Divide[s,\[Rho]])* Exp[s/ \[Rho]], {\[Rho], - Infinity, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [5 / 5]<div class="mw-collapsible-content"><code>{Plus[Complex[-0.02548520188983307, 3.121037092000815*^-18], Times[0.02420092827533985, NProduct[Times[Power[E, Times[-1.5, Power[ρ, -1]]], Plus[1, Times[1.5, Power[ρ, -1]]]] <- {ρ, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, -1.5]}</code><br><code>Plus[2.612375348685488, Times[-2.4801151038890965, NProduct[Times[Power[E, Times[1.5, Power[ρ, -1]]], Plus[1, Times[-1.5, Power[ρ, -1]]]] <- {ρ, DirectedInfinity[-1], DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[s, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.4.E1 25.4.E1] || [[Item:Q7608|<math>\Riemannzeta@{1-s} = 2(2\pi)^{-s}\cos@{\tfrac{1}{2}\pi s}\EulerGamma@{s}\Riemannzeta@{s}</math>]] || <code>Zeta(1 - s) = 2*(2*Pi)^(- s)* cos((1)/(2)*Pi*s)*GAMMA(s)*Zeta(s)</code> || <code>Zeta[1 - s] == 2*(2*Pi)^(- s)* Cos[Divide[1,2]*Pi*s]*Gamma[s]*Zeta[s]</code> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/25.4.E2 25.4.E2] || [[Item:Q7609|<math>\Riemannzeta@{s} = 2(2\pi)^{s-1}\sin@{\tfrac{1}{2}\pi s}\EulerGamma@{1-s}\Riemannzeta@{1-s}</math>]] || <code>Zeta(s) = 2*(2*Pi)^(s - 1)* sin((1)/(2)*Pi*s)*GAMMA(1 - s)*Zeta(1 - s)</code> || <code>Zeta[s] == 2*(2*Pi)^(s - 1)* Sin[Divide[1,2]*Pi*s]*Gamma[1 - s]*Zeta[1 - s]</code> || Failure || Successful || Successful [Tested: 4] || Successful [Tested: 4]
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| | [https://dlmf.nist.gov/25.4.E3 25.4.E3] || [[Item:Q7610|<math>\Riemannxi@{s} = \Riemannxi@{1-s}</math>]] || <code>(s)*(s-1)*GAMMA((s)/2)*Pi^(-(s)/2)*Zeta(s)/2 = (1 - s)*(1 - s-1)*GAMMA((1 - s)/2)*Pi^(-(1 - s)/2)*Zeta(1 - s)/2</code> || <code>RiemannXi[s] == RiemannXi[1 - s]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 6]<div class="mw-collapsible-content"><code>1/6]: [[Float(undefined)+Float(undefined)*I <- {s = -2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 6]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[s, -2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.4.E4 25.4.E4] || [[Item:Q7611|<math>\Riemannxi@{s} = \tfrac{1}{2}s(s-1)\EulerGamma@{\tfrac{1}{2}s}\pi^{-s/2}\Riemannzeta@{s}</math>]] || <code>(s)*(s-1)*GAMMA((s)/2)*Pi^(-(s)/2)*Zeta(s)/2 = (1)/(2)*s*(s - 1)* GAMMA((1)/(2)*s)*(Pi)^(- s/ 2)* Zeta(s)</code> || <code>RiemannXi[s] == Divide[1,2]*s*(s - 1)* Gamma[Divide[1,2]*s]*(Pi)^(- s/ 2)* Zeta[s]</code> || Successful || Successful || - || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/25.4.E5 25.4.E5] || [[Item:Q7612|<math>(-1)^{k}\Riemannzeta^{(k)}@{1-s} = \frac{2}{(2\pi)^{s}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{k}{m}\binom{m}{r}\left(\realpart@{c^{k-m}}\cos@{\tfrac{1}{2}\pi s}+\imagpart@{c^{k-m}}\sin@{\tfrac{1}{2}\pi s}\right)\EulerGamma^{(r)}@{s}\Riemannzeta^{(m-r)}@{s}</math>]] || <code>(- 1)^(k)* subs( temp=1 - s, diff( Zeta(temp), temp$(k) ) ) = (2)/((2*Pi)^(s))*sum(sum(binomial(k,m)*binomial(m,r)*(Re((c)^(k - m))*cos((1)/(2)*Pi*s)+ Im((c)^(k - m))*sin((1)/(2)*Pi*s))* diff( GAMMA(s), s$(r) )*diff( Zeta(s), s$(m - r) ), r = 0..m), m = 0..k)</code> || <code>(- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> 1 - s) == Divide[2,(2*Pi)^(s)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*(Re[(c)^(k - m)]*Cos[Divide[1,2]*Pi*s]+ Im[(c)^(k - m)]*Sin[Divide[1,2]*Pi*s])* D[Gamma[s], {s, r}]*D[Zeta[s], {s, m - r}], {r, 0, m}, GenerateConditions->None], {m, 0, k}, GenerateConditions->None]</code> || Aborted || Failure || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/25.5.E1 25.5.E1] || [[Item:Q7614|<math>\Riemannzeta@{s} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}\diff{x}</math>]] || <code>Zeta(s) = (1)/(GAMMA(s))*int(((x)^(s - 1))/(exp(x)- 1), x = 0..infinity)</code> || <code>Zeta[s] == Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]- 1], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 2] || Successful [Tested: 2]
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| | [https://dlmf.nist.gov/25.5.E2 25.5.E2] || [[Item:Q7615|<math>\Riemannzeta@{s} = \frac{1}{\EulerGamma@{s+1}}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}-1)^{2}}\diff{x}</math>]] || <code>Zeta(s) = (1)/(GAMMA(s + 1))*int((exp(x)*(x)^(s))/((exp(x)- 1)^(2)), x = 0..infinity)</code> || <code>Zeta[s] == Divide[1,Gamma[s + 1]]*Integrate[Divide[Exp[x]*(x)^(s),(Exp[x]- 1)^(2)], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || Successful [Tested: 2] || Successful [Tested: 2]
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| | [https://dlmf.nist.gov/25.5.E3 25.5.E3] || [[Item:Q7616|<math>\Riemannzeta@{s} = \frac{1}{(1-2^{1-s})\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}+1}\diff{x}</math>]] || <code>Zeta(s) = (1)/((1 - (2)^(1 - s))* GAMMA(s))*int(((x)^(s - 1))/(exp(x)+ 1), x = 0..infinity)</code> || <code>Zeta[s] == Divide[1,(1 - (2)^(1 - s))* Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]+ 1], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/25.5.E4 25.5.E4] || [[Item:Q7617|<math>\Riemannzeta@{s} = \frac{1}{(1-2^{1-s})\EulerGamma@{s+1}}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}+1)^{2}}\diff{x}</math>]] || <code>Zeta(s) = (1)/((1 - (2)^(1 - s))* GAMMA(s + 1))*int((exp(x)*(x)^(s))/((exp(x)+ 1)^(2)), x = 0..infinity)</code> || <code>Zeta[s] == Divide[1,(1 - (2)^(1 - s))* Gamma[s + 1]]*Integrate[Divide[Exp[x]*(x)^(s),(Exp[x]+ 1)^(2)], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/25.5.E5 25.5.E5] || [[Item:Q7618|<math>\Riemannzeta@{s} = -s\int_{0}^{\infty}\frac{x-\floor{x}-\frac{1}{2}}{x^{s+1}}\diff{x}</math>]] || <code>Zeta(s) = - s*int((x - floor(x)-(1)/(2))/((x)^(s + 1)), x = 0..infinity)</code> || <code>Zeta[s] == - s*Integrate[Divide[x - Floor[x]-Divide[1,2],(x)^(s + 1)], {x, 0, Infinity}, GenerateConditions->None]</code> || Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><code>{Complex[-2500601.4984594644, 2.5458720000534374*^-17] <- {Rule[s, -0.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.5.E6 25.5.E6] || [[Item:Q7619|<math>\Riemannzeta@{s} = \frac{1}{2}+\frac{1}{s-1}+\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{x}}\diff{x}</math>]] || <code>Zeta(s) = (1)/(2)+(1)/(s - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2))*((x)^(s - 1))/(exp(x)), x = 0..infinity)</code> || <code>Zeta[s] == Divide[1,2]+Divide[1,s - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2])*Divide[(x)^(s - 1),Exp[x]], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 3] || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 3]<div class="mw-collapsible-content"><code>{DirectedInfinity[] <- {Rule[s, 1.5]}</code><br><code>Indeterminate <- {Rule[s, 0.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.5.E7 25.5.E7] || [[Item:Q7620|<math>\Riemannzeta@{s} = \frac{1}{2}+\frac{1}{s-1}+\sum_{m=1}^{n}\frac{\BernoullinumberB{2m}}{(2m)!}\Pochhammersym{s}{2m-1}+\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac{\BernoullinumberB{2m}}{(2m)!}x^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\diff{x}</math>]] || <code>Zeta(s) = (1)/(2)+(1)/(s - 1)+ sum((bernoulli(2*m))/(factorial(2*m))*pochhammer(s, 2*m - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2)- sum((bernoulli(2*m))/(factorial(2*m))*(x)^(2*m - 1), m = 1..n))*((x)^(s - 1))/(exp(x)), x = 0..infinity), m = 1..n)</code> || <code>Zeta[s] == Divide[1,2]+Divide[1,s - 1]+ Sum[Divide[BernoulliB[2*m],(2*m)!]*Pochhammer[s, 2*m - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2]- Sum[Divide[BernoulliB[2*m],(2*m)!]*(x)^(2*m - 1), {m, 1, n}, GenerateConditions->None])*Divide[(x)^(s - 1),Exp[x]], {x, 0, Infinity}, GenerateConditions->None], {m, 1, n}, GenerateConditions->None]</code> || Aborted || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><code>2/3]: [[.1027534444e-1 <- {s = 3/2, n = 3}</code><br><code>.24417579e-1 <- {s = 2, n = 3}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/25.5.E8 25.5.E8] || [[Item:Q7621|<math>\Riemannzeta@{s} = \frac{1}{2(1-2^{-s})\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{\sinh@@{x}}\diff{x}</math>]] || <code>Zeta(s) = (1)/(2*(1 - (2)^(- s))* GAMMA(s))*int(((x)^(s - 1))/(sinh(x)), x = 0..infinity)</code> || <code>Zeta[s] == Divide[1,2*(1 - (2)^(- s))* Gamma[s]]*Integrate[Divide[(x)^(s - 1),Sinh[x]], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 2] || Successful [Tested: 2]
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| | [https://dlmf.nist.gov/25.5.E9 25.5.E9] || [[Item:Q7622|<math>\Riemannzeta@{s} = \frac{2^{s-1}}{\EulerGamma@{s+1}}\int_{0}^{\infty}\frac{x^{s}}{(\sinh@@{x})^{2}}\diff{x}</math>]] || <code>Zeta(s) = ((2)^(s - 1))/(GAMMA(s + 1))*int(((x)^(s))/((sinh(x))^(2)), x = 0..infinity)</code> || <code>Zeta[s] == Divide[(2)^(s - 1),Gamma[s + 1]]*Integrate[Divide[(x)^(s),(Sinh[x])^(2)], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || Successful [Tested: 2] || Skipped - Because timed out
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| | [https://dlmf.nist.gov/25.5.E10 25.5.E10] || [[Item:Q7623|<math>\Riemannzeta@{s} = \frac{2^{s-1}}{1-2^{1-s}}\int_{0}^{\infty}\frac{\cos@{s\atan@@{x}}}{(1+x^{2})^{s/2}\cosh@{\frac{1}{2}\pi x}}\diff{x}</math>]] || <code>Zeta(s) = ((2)^(s - 1))/(1 - (2)^(1 - s))*int((cos(s*arctan(x)))/((1 + (x)^(2))^(s/ 2)* cosh((1)/(2)*Pi*x)), x = 0..infinity)</code> || <code>Zeta[s] == Divide[(2)^(s - 1),1 - (2)^(1 - s)]*Integrate[Divide[Cos[s*ArcTan[x]],(1 + (x)^(2))^(s/ 2)* Cosh[Divide[1,2]*Pi*x]], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || Successful [Tested: 6] || Skipped - Because timed out
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| | [https://dlmf.nist.gov/25.5.E11 25.5.E11] || [[Item:Q7624|<math>\Riemannzeta@{s} = \frac{1}{2}+\frac{1}{s-1}+2\int_{0}^{\infty}\frac{\sin@{s\atan@@{x}}}{(1+x^{2})^{s/2}(e^{2\pi x}-1)}\diff{x}</math>]] || <code>Zeta(s) = (1)/(2)+(1)/(s - 1)+ 2*int((sin(s*arctan(x)))/((1 + (x)^(2))^(s/ 2)*(exp(2*Pi*x)- 1)), x = 0..infinity)</code> || <code>Zeta[s] == Divide[1,2]+Divide[1,s - 1]+ 2*Integrate[Divide[Sin[s*ArcTan[x]],(1 + (x)^(2))^(s/ 2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 6] || Successful [Tested: 6]
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| | [https://dlmf.nist.gov/25.5.E12 25.5.E12] || [[Item:Q7625|<math>\Riemannzeta@{s} = \frac{2^{s-1}}{s-1}-2^{s}\int_{0}^{\infty}\frac{\sin@{s\atan@@{x}}}{(1+x^{2})^{s/2}(e^{\pi x}+1)}\diff{x}</math>]] || <code>Zeta(s) = ((2)^(s - 1))/(s - 1)- (2)^(s)* int((sin(s*arctan(x)))/((1 + (x)^(2))^(s/ 2)*(exp(Pi*x)+ 1)), x = 0..infinity)</code> || <code>Zeta[s] == Divide[(2)^(s - 1),s - 1]- (2)^(s)* Integrate[Divide[Sin[s*ArcTan[x]],(1 + (x)^(2))^(s/ 2)*(Exp[Pi*x]+ 1)], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 6] || Successful [Tested: 6]
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| |-
| |
| | [https://dlmf.nist.gov/25.5.E13 25.5.E13] || [[Item:Q7626|<math>\Riemannzeta@{s} = \frac{\pi^{s/2}}{s(s-1)\EulerGamma@{\frac{1}{2}s}}+\frac{\pi^{s/2}}{\EulerGamma@{\frac{1}{2}s}}\*\int_{1}^{\infty}\left(x^{s/2}+x^{(1-s)/2}\right)\frac{\omega(x)}{x}\diff{x}</math>]] || <code>Zeta(s) = ((Pi)^(s/ 2))/(s*(s - 1)* GAMMA((1)/(2)*s))+((Pi)^(s/ 2))/(GAMMA((1)/(2)*s))* int(((x)^(s/ 2)+ (x)^((1 - s)/ 2))*(omega*(x))/(x), x = 1..infinity)</code> || <code>Zeta[s] == Divide[(Pi)^(s/ 2),s*(s - 1)* Gamma[Divide[1,2]*s]]+Divide[(Pi)^(s/ 2),Gamma[Divide[1,2]*s]]* Integrate[((x)^(s/ 2)+ (x)^((1 - s)/ 2))*Divide[\[Omega]*(x),x], {x, 1, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><code>30/30]: [[Float(infinity)+Float(infinity)*I <- {omega = 1/2*3^(1/2)+1/2*I, s = 3/2}</code><br><code>Float(infinity)+Float(infinity)*I <- {omega = 1/2*3^(1/2)+1/2*I, s = 1/2}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/25.5.E14 25.5.E14] || [[Item:Q7627|<math>\omega(x)\defeq\sum_{n=1}^{\infty}e^{-n^{2}\pi x} = \frac{1}{2}\left(\Jacobithetatau{3}@{0}{ix}-1\right)</math>]] || <code>omega*(x) = sum(exp(- (n)^(2)* Pi*x), n = 1..infinity) = (1)/(2)*(JacobiTheta3(0,exp(I*Pi*I*x))- 1)</code> || <code>\[Omega]*(x) == Sum[Exp[- (n)^(2)* Pi*x], {n, 1, Infinity}, GenerateConditions->None] == Divide[1,2]*(EllipticTheta[3, 0, Exp[I*Pi*(I*x)]]- 1)</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><code>{Plus[-0.008983297533541545, False] <- {Rule[x, 1.5], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[-0.008983297533541545, False] <- {Rule[x, 1.5], Rule[ω, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/25.5.E15 25.5.E15] || [[Item:Q7628|<math>\Riemannzeta@{s} = \frac{1}{s-1}+\frac{\sin@{\pi s}}{\pi}\*\int_{0}^{\infty}(\ln@{1+x}-\digamma@{1+x})x^{-s}\diff{x}</math>]] || <code>Zeta(s) = (1)/(s - 1)+(sin(Pi*s))/(Pi)* int((ln(1 + x)- Psi(1 + x))* (x)^(- s), x = 0..infinity)</code> || <code>Zeta[s] == Divide[1,s - 1]+Divide[Sin[Pi*s],Pi]* Integrate[(Log[1 + x]- PolyGamma[1 + x])* (x)^(- s), {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [5 / 6]<div class="mw-collapsible-content"><code>5/6]: [[Float(-infinity) <- {s = -3/2}</code><br><code>Float(infinity) <- {s = 3/2}</code><br></div></div> || Skipped - Because timed out
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| |-
| |
| | [https://dlmf.nist.gov/25.5.E16 25.5.E16] || [[Item:Q7629|<math>\Riemannzeta@{s} = \frac{1}{s-1}+\frac{\sin@{\pi s}}{\pi(s-1)}\*\int_{0}^{\infty}\left(\frac{1}{1+x}-\digamma'@{1+x}\right)x^{1-s}\diff{x}</math>]] || <code>Zeta(s) = (1)/(s - 1)+(sin(Pi*s))/(Pi*(s - 1))* int(((1)/(1 + x)- subs( temp=1 + x, diff( Psi(temp), temp$(1) ) ))* (x)^(1 - s), x = 0..infinity)</code> || <code>Zeta[s] == Divide[1,s - 1]+Divide[Sin[Pi*s],Pi*(s - 1)]* Integrate[(Divide[1,1 + x]- (D[PolyGamma[temp], {temp, 1}]/.temp-> 1 + x))* (x)^(1 - s), {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [4 / 6]<div class="mw-collapsible-content"><code>4/6]: [[Float(-infinity) <- {s = -3/2}</code><br><code>Float(infinity) <- {s = -1/2}</code><br></div></div> || Skipped - Because timed out
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| |-
| |
| | [https://dlmf.nist.gov/25.5.E17 25.5.E17] || [[Item:Q7630|<math>\Riemannzeta@{1+s} = \frac{\sin@{\pi s}}{\pi}\int_{0}^{\infty}\left(\EulerConstant+\digamma@{1+x}\right)x^{-s-1}\diff{x}</math>]] || <code>Zeta(1 + s) = (sin(Pi*s))/(Pi)*int((gamma + Psi(1 + x))* (x)^(- s - 1), x = 0..infinity)</code> || <code>Zeta[1 + s] == Divide[Sin[Pi*s],Pi]*Integrate[(EulerGamma + PolyGamma[1 + x])* (x)^(- s - 1), {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 6]<div class="mw-collapsible-content"><code>6/6]: [[Float(-infinity) <- {s = -3/2}</code><br><code>Float(-infinity) <- {s = 3/2}</code><br></div></div> || Skipped - Because timed out
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| |
| | [https://dlmf.nist.gov/25.5.E18 25.5.E18] || [[Item:Q7631|<math>\Riemannzeta@{1+s} = \frac{\sin@{\pi s}}{\pi s}\int_{0}^{\infty}\digamma'@{1+x}x^{-s}\diff{x}</math>]] || <code>Zeta(1 + s) = (sin(Pi*s))/(Pi*s)*int(subs( temp=1 + x, diff( Psi(temp), temp$(1) ) )*(x)^(- s), x = 0..infinity)</code> || <code>Zeta[1 + s] == Divide[Sin[Pi*s],Pi*s]*Integrate[(D[PolyGamma[temp], {temp, 1}]/.temp-> 1 + x)*(x)^(- s), {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [5 / 6]<div class="mw-collapsible-content"><code>5/6]: [[Float(infinity) <- {s = -3/2}</code><br><code>Float(infinity) <- {s = 3/2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [5 / 6]<div class="mw-collapsible-content"><code>{Complex[1.225514223589735984806690657246494542888138*^+10484, 2.5458720000534374*^-17] <- {Rule[s, -1.5]}</code><br><code>2.9996624622276097*^37 <- {Rule[s, 1.5]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.5.E19 25.5.E19] || [[Item:Q7632|<math>\Riemannzeta@{m+s} = (-1)^{m-1}\frac{\EulerGamma@{s}\sin@{\pi s}}{\pi\EulerGamma@{m+s}}\*\int_{0}^{\infty}\digamma^{(m)}@{1+x}x^{-s}\diff{x}</math>]] || <code>Zeta(m + s) = (- 1)^(m - 1)*(GAMMA(s)*sin(Pi*s))/(Pi*GAMMA(m + s))* int(subs( temp=1 + x, diff( Psi(temp), temp$(m) ) )*(x)^(- s), x = 0..infinity)</code> || <code>Zeta[m + s] == (- 1)^(m - 1)*Divide[Gamma[s]*Sin[Pi*s],Pi*Gamma[m + s]]* Integrate[(D[PolyGamma[temp], {temp, m}]/.temp-> 1 + x)*(x)^(- s), {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 3]<div class="mw-collapsible-content"><code>2/3]: [[Float(infinity) <- {s = 3/2, m = 3}</code><br><code>Float(-infinity) <- {s = 2, m = 3}</code><br></div></div> || Skipped - Because timed out
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| |
| | [https://dlmf.nist.gov/25.5.E20 25.5.E20] || [[Item:Q7633|<math>\Riemannzeta@{s} = \frac{\EulerGamma@{1-s}}{2\pi i}\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}-1}\diff{z}</math>]] || <code>Zeta(s) = (GAMMA(1 - s))/(2*Pi*I)*int(((z)^(s - 1))/(exp(- z)- 1), z = - infinity..(0 +))</code> || <code>Zeta[s] == Divide[Gamma[1 - s],2*Pi*I]*Integrate[Divide[(z)^(s - 1),Exp[- z]- 1], {z, - Infinity, (0 +)}, GenerateConditions->None]</code> || Error || Failure || - || Error
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| |-
| |
| | [https://dlmf.nist.gov/25.5.E21 25.5.E21] || [[Item:Q7634|<math>\Riemannzeta@{s} = \frac{\EulerGamma@{1-s}}{2\pi i(1-2^{1-s})}\*\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}+1}\diff{z}</math>]] || <code>Zeta(s) = (GAMMA(1 - s))/(2*Pi*I*(1 - (2)^(1 - s)))* int(((z)^(s - 1))/(exp(- z)+ 1), z = - infinity..(0 +))</code> || <code>Zeta[s] == Divide[Gamma[1 - s],2*Pi*I*(1 - (2)^(1 - s))]* Integrate[Divide[(z)^(s - 1),Exp[- z]+ 1], {z, - Infinity, (0 +)}, GenerateConditions->None]</code> || Error || Failure || - || Error
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| |-
| |
| | [https://dlmf.nist.gov/25.6#Ex1 25.6#Ex1] || [[Item:Q7635|<math>\Riemannzeta@{0} = -\frac{1}{2}</math>]] || <code>Zeta(0) = -(1)/(2)</code> || <code>Zeta[0] == -Divide[1,2]</code> || Successful || Successful || - || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.6#Ex2 25.6#Ex2] || [[Item:Q7636|<math>\Riemannzeta@{2} = \frac{\pi^{2}}{6}</math>]] || <code>Zeta(2) = ((Pi)^(2))/(6)</code> || <code>Zeta[2] == Divide[(Pi)^(2),6]</code> || Successful || Successful || - || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.6#Ex3 25.6#Ex3] || [[Item:Q7637|<math>\Riemannzeta@{4} = \frac{\pi^{4}}{90}</math>]] || <code>Zeta(4) = ((Pi)^(4))/(90)</code> || <code>Zeta[4] == Divide[(Pi)^(4),90]</code> || Successful || Successful || - || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.6#Ex4 25.6#Ex4] || [[Item:Q7638|<math>\Riemannzeta@{6} = \frac{\pi^{6}}{945}</math>]] || <code>Zeta(6) = ((Pi)^(6))/(945)</code> || <code>Zeta[6] == Divide[(Pi)^(6),945]</code> || Successful || Successful || - || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.6.E2 25.6.E2] || [[Item:Q7639|<math>\Riemannzeta@{2n} = \frac{(2\pi)^{2n}}{2(2n)!}\left|\BernoullinumberB{2n}\right|</math>]] || <code>Zeta(2*n) = ((2*Pi)^(2*n))/(2*factorial(2*n))*abs(bernoulli(2*n))</code> || <code>Zeta[2*n] == Divide[(2*Pi)^(2*n),2*(2*n)!]*Abs[BernoulliB[2*n]]</code> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.6.E3 25.6.E3] || [[Item:Q7640|<math>\Riemannzeta@{-n} = -\frac{\BernoullinumberB{n+1}}{n+1}</math>]] || <code>Zeta(- n) = -(bernoulli(n + 1))/(n + 1)</code> || <code>Zeta[- n] == -Divide[BernoulliB[n + 1],n + 1]</code> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.6.E4 25.6.E4] || [[Item:Q7641|<math>\Riemannzeta@{-2n} = 0</math>]] || <code>Zeta(- 2*n) = 0</code> || <code>Zeta[- 2*n] == 0</code> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.6.E6 25.6.E6] || [[Item:Q7643|<math>\Riemannzeta@{2k+1} = \frac{(-1)^{k+1}(2\pi)^{2k+1}}{2(2k+1)!}\int_{0}^{1}\BernoullipolyB{2k+1}@{t}\cot@{\pi t}\diff{t}</math>]] || <code>Zeta(2*k + 1) = ((- 1)^(k + 1)*(2*Pi)^(2*k + 1))/(2*factorial(2*k + 1))*int(bernoulli(2*k + 1, t)*cot(Pi*t), t = 0..1)</code> || <code>Zeta[2*k + 1] == Divide[(- 1)^(k + 1)*(2*Pi)^(2*k + 1),2*(2*k + 1)!]*Integrate[BernoulliB[2*k + 1, t]*Cot[Pi*t], {t, 0, 1}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.6.E7 25.6.E7] || [[Item:Q7644|<math>\Riemannzeta@{2} = \int_{0}^{1}\int_{0}^{1}\frac{1}{1-xy}\diff{x}\diff{y}</math>]] || <code>Zeta(2) = int(int((1)/(1 - x*y), x = 0..1), y = 0..1)</code> || <code>Zeta[2] == Integrate[Integrate[Divide[1,1 - x*y], {x, 0, 1}, GenerateConditions->None], {y, 0, 1}, GenerateConditions->None]</code> || Successful || Successful || - || Successful [Tested: 1]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.6.E8 25.6.E8] || [[Item:Q7645|<math>\Riemannzeta@{2} = 3\sum_{k=1}^{\infty}\frac{1}{k^{2}\binom{2k}{k}}</math>]] || <code>Zeta(2) = 3*sum((1)/((k)^(2)*binomial(2*k,k)), k = 1..infinity)</code> || <code>Zeta[2] == 3*Sum[Divide[1,(k)^(2)*Binomial[2*k,k]], {k, 1, Infinity}, GenerateConditions->None]</code> || Successful || Successful || - || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.6.E9 25.6.E9] || [[Item:Q7646|<math>\Riemannzeta@{3} = \frac{5}{2}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k^{3}\binom{2k}{k}}</math>]] || <code>Zeta(3) = (5)/(2)*sum(((- 1)^(k - 1))/((k)^(3)*binomial(2*k,k)), k = 1..infinity)</code> || <code>Zeta[3] == Divide[5,2]*Sum[Divide[(- 1)^(k - 1),(k)^(3)*Binomial[2*k,k]], {k, 1, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.6.E10 25.6.E10] || [[Item:Q7647|<math>\Riemannzeta@{4} = \frac{36}{17}\sum_{k=1}^{\infty}\frac{1}{k^{4}\binom{2k}{k}}</math>]] || <code>Zeta(4) = (36)/(17)*sum((1)/((k)^(4)*binomial(2*k,k)), k = 1..infinity)</code> || <code>Zeta[4] == Divide[36,17]*Sum[Divide[1,(k)^(4)*Binomial[2*k,k]], {k, 1, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.6.E11 25.6.E11] || [[Item:Q7648|<math>\Riemannzeta'@{0} = -\tfrac{1}{2}\ln@{2\pi}</math>]] || <code>subs( temp=0, diff( Zeta(temp), temp$(1) ) ) = -(1)/(2)*ln(2*Pi)</code> || <code>(D[Zeta[temp], {temp, 1}]/.temp-> 0) == -Divide[1,2]*Log[2*Pi]</code> || Successful || Successful || - || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.6.E12 25.6.E12] || [[Item:Q7649|<math>\Riemannzeta''@{0} = -\tfrac{1}{2}(\ln@{2\pi})^{2}+\tfrac{1}{2}\EulerConstant^{2}-\tfrac{1}{24}\pi^{2}+\StieltjesConstants{1}</math>]] || <code>subs( temp=0, diff( Zeta(temp), temp$(2) ) ) = -(1)/(2)*(ln(2*Pi))^(2)+(1)/(2)*(gamma)^(2)-(1)/(24)*(Pi)^(2)+ gamma(1)</code> || <code>Error</code> || Successful || Missing Macro Error || - || -
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| |-
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| | [https://dlmf.nist.gov/25.6.E13 25.6.E13] || [[Item:Q7650|<math>(-1)^{k}\Riemannzeta^{(k)}@{-2n} = \frac{2(-1)^{n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{k}{m}\binom{m}{r}\imagpart@{c^{k-m}}\*\EulerGamma^{(r)}@{2n+1}\Riemannzeta^{(m-r)}@{2n+1}</math>]] || <code>(- 1)^(k)* subs( temp=- 2*n, diff( Zeta(temp), temp$(k) ) ) = (2*(- 1)^(n))/((2*Pi)^(2*n + 1))*sum(sum(binomial(k,m)*binomial(m,r)*Im((c)^(k - m))* subs( temp=2*n + 1, diff( GAMMA(temp), temp$(r) ) )*subs( temp=2*n + 1, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k)</code> || <code>(- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> - 2*n) == Divide[2*(- 1)^(n),(2*Pi)^(2*n + 1)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*Im[(c)^(k - m)]* (D[Gamma[temp], {temp, r}]/.temp-> 2*n + 1)*(D[Zeta[temp], {temp, m - r}]/.temp-> 2*n + 1), {r, 0, m}, GenerateConditions->None], {m, 0, k}, GenerateConditions->None]</code> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [48 / 54]<div class="mw-collapsible-content"><code>{0.030448457058393275 <- {Rule[c, -1.5], Rule[k, 1], Rule[n, 1]}</code><br><code>-0.007983811450268627 <- {Rule[c, -1.5], Rule[k, 1], Rule[n, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.6.E14 25.6.E14] || [[Item:Q7651|<math>(-1)^{k}\Riemannzeta^{(k)}@{1-2n} = \frac{2(-1)^{n}}{(2\pi)^{2n}}\sum_{m=0}^{k}\sum_{r=0}^{m}\binom{k}{m}\binom{m}{r}\realpart@{c^{k-m}}\*\EulerGamma^{(r)}@{2n}\Riemannzeta^{(m-r)}@{2n}</math>]] || <code>(- 1)^(k)* subs( temp=1 - 2*n, diff( Zeta(temp), temp$(k) ) ) = (2*(- 1)^(n))/((2*Pi)^(2*n))*sum(sum(binomial(k,m)*binomial(m,r)*Re((c)^(k - m))* subs( temp=2*n, diff( GAMMA(temp), temp$(r) ) )*subs( temp=2*n, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k)</code> || <code>(- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> 1 - 2*n) == Divide[2*(- 1)^(n),(2*Pi)^(2*n)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*Re[(c)^(k - m)]* (D[Gamma[temp], {temp, r}]/.temp-> 2*n)*(D[Zeta[temp], {temp, m - r}]/.temp-> 2*n), {r, 0, m}, GenerateConditions->None], {m, 0, k}, GenerateConditions->None]</code> || Aborted || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [54 / 54]<div class="mw-collapsible-content"><code>{Plus[0.1654211437004509, Times[0.05066059182116889, Plus[Times[-1.0772156649015328, D[1.6449340668482262 <- {2.0, 0.0}]], Times[1.0, D[1.6449340668482262, {2.0, 1.0}]]]]], {Rule[c, -1.5], Rule[k, 1], Rule[n, 1]}</code><br><code>Plus[-0.005378576357774297, Times[-0.001283247781835542, Plus[Times[-1.4632939894091983, D[1.0823232337111381 <- {4.0, 0.0}]], Times[6.0, D[1.0823232337111381, {4.0, 1.0}]]]]], {Rule[c, -1.5], Rule[k, 1], Rule[n, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.6.E15 25.6.E15] || [[Item:Q7652|<math>\Riemannzeta'@{2n} = \frac{(-1)^{n+1}(2\pi)^{2n}}{2(2n)!}\left(2n\Riemannzeta'@{1-2n}-(\digamma@{2n}-\ln@{2\pi})\BernoullinumberB{2n}\right)</math>]] || <code>subs( temp=2*n, diff( Zeta(temp), temp$(1) ) ) = ((- 1)^(n + 1)*(2*Pi)^(2*n))/(2*factorial(2*n))*(2*n*subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) )-(Psi(2*n)- ln(2*Pi))*bernoulli(2*n))</code> || <code>(D[Zeta[temp], {temp, 1}]/.temp-> 2*n) == Divide[(- 1)^(n + 1)*(2*Pi)^(2*n),2*(2*n)!]*(2*n*(D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n)-(PolyGamma[2*n]- Log[2*Pi])*BernoulliB[2*n])</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| |-
| |
| | [https://dlmf.nist.gov/25.6.E16 25.6.E16] || [[Item:Q7653|<math>\left(n+\tfrac{1}{2}\right)\Riemannzeta@{2n} = \sum_{k=1}^{n-1}\Riemannzeta@{2k}\Riemannzeta@{2n-2k}</math>]] || <code>(n +(1)/(2))* Zeta(2*n) = sum(Zeta(2*k)*Zeta(2*n - 2*k), k = 1..n - 1)</code> || <code>(n +Divide[1,2])* Zeta[2*n] == Sum[Zeta[2*k]*Zeta[2*n - 2*k], {k, 1, n - 1}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
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| | [https://dlmf.nist.gov/25.6.E17 25.6.E17] || [[Item:Q7654|<math>\left(n+\tfrac{3}{4}\right)\Riemannzeta@{4n+2} = \sum_{k=1}^{n}\Riemannzeta@{2k}\Riemannzeta@{4n+2-2k}</math>]] || <code>(n +(3)/(4))* Zeta(4*n + 2) = sum(Zeta(2*k)*Zeta(4*n + 2 - 2*k), k = 1..n)</code> || <code>(n +Divide[3,4])* Zeta[4*n + 2] == Sum[Zeta[2*k]*Zeta[4*n + 2 - 2*k], {k, 1, n}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/25.6.E18 25.6.E18] || [[Item:Q7655|<math>\left(n+\tfrac{1}{4}\right)\Riemannzeta@{4n}+\tfrac{1}{2}(\Riemannzeta@{2n})^{2} = \sum_{k=1}^{n}\Riemannzeta@{2k}\Riemannzeta@{4n-2k}</math>]] || <code>(n +(1)/(4))* Zeta(4*n)+(1)/(2)*(Zeta(2*n))^(2) = sum(Zeta(2*k)*Zeta(4*n - 2*k), k = 1..n)</code> || <code>(n +Divide[1,4])* Zeta[4*n]+Divide[1,2]*(Zeta[2*n])^(2) == Sum[Zeta[2*k]*Zeta[4*n - 2*k], {k, 1, n}, GenerateConditions->None]</code> || Skipped - Unable to analyze test case: Null || Skipped - Unable to analyze test case: Null || - || -
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| |-
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| | [https://dlmf.nist.gov/25.6.E20 25.6.E20] || [[Item:Q7657|<math>\tfrac{1}{2}(2^{2n}-1)\Riemannzeta@{2n} = \sum_{k=1}^{n-1}(2^{2n-2k}-1)\Riemannzeta@{2n-2k}\Riemannzeta@{2k}</math>]] || <code>(1)/(2)*((2)^(2*n)- 1)* Zeta(2*n) = sum(((2)^(2*n - 2*k)- 1)* Zeta(2*n - 2*k)*Zeta(2*k), k = 1..n - 1)</code> || <code>Divide[1,2]*((2)^(2*n)- 1)* Zeta[2*n] == Sum[((2)^(2*n - 2*k)- 1)* Zeta[2*n - 2*k]*Zeta[2*k], {k, 1, n - 1}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 2] || Successful [Tested: 2]
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| |-
| |
| | [https://dlmf.nist.gov/25.8.E1 25.8.E1] || [[Item:Q7658|<math>\sum_{k=2}^{\infty}\left(\Riemannzeta@{k}-1\right) = 1</math>]] || <code>sum(Zeta(k)- 1, k = 2..infinity) = 1</code> || <code>Sum[Zeta[k]- 1, {k, 2, Infinity}, GenerateConditions->None] == 1</code> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
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| |
| | [https://dlmf.nist.gov/25.8.E2 25.8.E2] || [[Item:Q7659|<math>\sum_{k=0}^{\infty}\frac{\EulerGamma@{s+k}}{(k+1)!}\left(\Riemannzeta@{s+k}-1\right) = \EulerGamma@{s-1}</math>]] || <code>sum((GAMMA(s + k))/(factorial(k + 1))*(Zeta(s + k)- 1), k = 0..infinity) = GAMMA(s - 1)</code> || <code>Sum[Divide[Gamma[s + k],(k + 1)!]*(Zeta[s + k]- 1), {k, 0, Infinity}, GenerateConditions->None] == Gamma[s - 1]</code> || Failure || Aborted || Successful [Tested: 1] || Skipped - Because timed out
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| |-
| |
| | [https://dlmf.nist.gov/25.8.E3 25.8.E3] || [[Item:Q7660|<math>\sum_{k=0}^{\infty}\frac{\Pochhammersym{s}{k}\Riemannzeta@{s+k}}{k!2^{s+k}} = (1-2^{-s})\Riemannzeta@{s}</math>]] || <code>sum((pochhammer(s, k)*Zeta(s + k))/(factorial(k)*(2)^(s + k)), k = 0..infinity) = (1 - (2)^(- s))* Zeta(s)</code> || <code>Sum[Divide[Pochhammer[s, k]*Zeta[s + k],(k)!*(2)^(s + k)], {k, 0, Infinity}, GenerateConditions->None] == (1 - (2)^(- s))* Zeta[s]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 6]<div class="mw-collapsible-content"><code>1/6]: [[-.1666666667 <- {s = -2}</code><br></div></div> || Successful [Tested: 6]
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| |-
| |
| | [https://dlmf.nist.gov/25.8.E4 25.8.E4] || [[Item:Q7661|<math>\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\Riemannzeta@{nk}-1) = \ln@{\prod_{j=0}^{n-1}\EulerGamma@{2-e^{(2j+1)\pi i/n}}}</math>]] || <code>sum(((- 1)^(k))/(k)*(Zeta(n*k)- 1), k = 1..infinity) = ln(product(GAMMA(2 - exp((2*j + 1)* Pi*I/ n)), j = 0..n - 1))</code> || <code>Sum[Divide[(- 1)^(k),k]*(Zeta[n*k]- 1), {k, 1, Infinity}, GenerateConditions->None] == Log[Product[Gamma[2 - Exp[(2*j + 1)* Pi*I/ n]], {j, 0, n - 1}, GenerateConditions->None]]</code> || Failure || Failure || Successful [Tested: 1] || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><code>{Plus[-0.6931471805599453, NSum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[k]]] <- {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[n, 1]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/25.8.E5 25.8.E5] || [[Item:Q7662|<math>\sum_{k=2}^{\infty}\Riemannzeta@{k}z^{k} = -\EulerConstant z-z\digamma@{1-z}</math>]] || <code>sum(Zeta(k)*(z)^(k), k = 2..infinity) = - gamma*z - z*Psi(1 - z)</code> || <code>Sum[Zeta[k]*(z)^(k), {k, 2, Infinity}, GenerateConditions->None] == - EulerGamma*z - z*PolyGamma[1 - z]</code> || Failure || Successful || Successful [Tested: 1] || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.8.E6 25.8.E6] || [[Item:Q7663|<math>\sum_{k=0}^{\infty}\Riemannzeta@{2k}z^{2k} = -\tfrac{1}{2}\pi z\cot@{\pi z}</math>]] || <code>sum(Zeta(2*k)*(z)^(2*k), k = 0..infinity) = -(1)/(2)*Pi*z*cot(Pi*z)</code> || <code>Sum[Zeta[2*k]*(z)^(2*k), {k, 0, Infinity}, GenerateConditions->None] == -Divide[1,2]*Pi*z*Cot[Pi*z]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><code>{Plus[4.8091767343044744*^-17, NSum[Times[Power[0.5, Times[2, k]], Zeta[Times[2, k]]] <- {k, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[z, 0.5]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/25.8.E7 25.8.E7] || [[Item:Q7664|<math>\sum_{k=2}^{\infty}\frac{\Riemannzeta@{k}}{k}z^{k} = -\EulerConstant z+\ln@@{\EulerGamma@{1-z}}</math>]] || <code>sum((Zeta(k))/(k)*(z)^(k), k = 2..infinity) = - gamma*z + ln(GAMMA(1 - z))</code> || <code>Sum[Divide[Zeta[k],k]*(z)^(k), {k, 2, Infinity}, GenerateConditions->None] == - EulerGamma*z + Log[Gamma[1 - z]]</code> || Failure || Successful || Successful [Tested: 1] || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.8.E8 25.8.E8] || [[Item:Q7665|<math>\sum_{k=1}^{\infty}\frac{\Riemannzeta@{2k}}{k}z^{2k} = \ln@{\frac{\pi z}{\sin@{\pi z}}}</math>]] || <code>sum((Zeta(2*k))/(k)*(z)^(2*k), k = 1..infinity) = ln((Pi*z)/(sin(Pi*z)))</code> || <code>Sum[Divide[Zeta[2*k],k]*(z)^(2*k), {k, 1, Infinity}, GenerateConditions->None] == Log[Divide[Pi*z,Sin[Pi*z]]]</code> || Failure || Successful || Successful [Tested: 1] || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.8.E9 25.8.E9] || [[Item:Q7666|<math>\sum_{k=1}^{\infty}\frac{\Riemannzeta@{2k}}{(2k+1)2^{2k}} = \frac{1}{2}-\frac{1}{2}\ln@@{2}</math>]] || <code>sum((Zeta(2*k))/((2*k + 1)* (2)^(2*k)), k = 1..infinity) = (1)/(2)-(1)/(2)*ln(2)</code> || <code>Sum[Divide[Zeta[2*k],(2*k + 1)* (2)^(2*k)], {k, 1, Infinity}, GenerateConditions->None] == Divide[1,2]-Divide[1,2]*Log[2]</code> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.8.E10 25.8.E10] || [[Item:Q7667|<math>\sum_{k=1}^{\infty}\frac{\Riemannzeta@{2k}}{(2k+1)(2k+2)2^{2k}} = \frac{1}{4}-\frac{7}{4\pi^{2}}\Riemannzeta@{3}</math>]] || <code>sum((Zeta(2*k))/((2*k + 1)*(2*k + 2)* (2)^(2*k)), k = 1..infinity) = (1)/(4)-(7)/(4*(Pi)^(2))*Zeta(3)</code> || <code>Sum[Divide[Zeta[2*k],(2*k + 1)*(2*k + 2)* (2)^(2*k)], {k, 1, Infinity}, GenerateConditions->None] == Divide[1,4]-Divide[7,4*(Pi)^(2)]*Zeta[3]</code> || Failure || Successful || Successful [Tested: 0] || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.10.E3 25.10.E3] || [[Item:Q7673|<math>Z(t) = 2\sum_{n=1}^{m}\frac{\cos@{\vartheta(t)-t\ln@@{n}}}{n^{1/2}}+R(t)</math>]] || <code>Z*(t) = 2*sum((cos(vartheta*(t)- t*ln(n)))/((n)^(1/ 2)), n = 1..m)+ R*(t)</code> || <code>Z*(t) == 2*Sum[Divide[Cos[\[CurlyTheta]*(t)- t*Log[n]],(n)^(1/ 2)], {n, 1, m}, GenerateConditions->None]+ R*(t)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>300/300]: [[-.6950521340+1.584276130*I <- {R = 1/2*3^(1/2)+1/2*I, Z = 1/2*3^(1/2)+1/2*I, t = -3/2, vartheta = 1/2*3^(1/2)+1/2*I, m = 1}</code><br><code>-2.464793153+1.882475916*I <- {R = 1/2*3^(1/2)+1/2*I, Z = 1/2*3^(1/2)+1/2*I, t = -3/2, vartheta = 1/2*3^(1/2)+1/2*I, m = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [300 / 300]<div class="mw-collapsible-content"><code>{Complex[-0.6950521348622749, 1.5842761296673835] <- {Rule[m, 1], Rule[R, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[t, -1.5], Rule[Z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϑ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-2.4647931552284597, 1.8824759153846262] <- {Rule[m, 2], Rule[R, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[t, -1.5], Rule[Z, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]], Rule[ϑ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| |
| | [https://dlmf.nist.gov/25.11.E1 25.11.E1] || [[Item:Q7675|<math>\Hurwitzzeta@{s}{a} = \sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}</math>]] || <code>Zeta(0, s, a) = sum((1)/((n + a)^(s)), n = 0..infinity)</code> || <code>HurwitzZeta[s, a] == Sum[Divide[1,(n + a)^(s)], {n, 0, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 2] || Successful [Tested: 2]
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| |
| | [https://dlmf.nist.gov/25.11.E2 25.11.E2] || [[Item:Q7676|<math>\Hurwitzzeta@{s}{1} = \Riemannzeta@{s}</math>]] || <code>Zeta(0, s, 1) = Zeta(s)</code> || <code>HurwitzZeta[s, 1] == Zeta[s]</code> || Successful || Successful || - || Successful [Tested: 6]
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| |-
| |
| | [https://dlmf.nist.gov/25.11.E3 25.11.E3] || [[Item:Q7677|<math>\Hurwitzzeta@{s}{a} = \Hurwitzzeta@{s}{a+1}+a^{-s}</math>]] || <code>Zeta(0, s, a) = Zeta(0, s, a + 1)+ (a)^(- s)</code> || <code>HurwitzZeta[s, a] == HurwitzZeta[s, a + 1]+ (a)^(- s)</code> || Failure || Successful || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [3 / 36]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, -2], Rule[s, 1.5]}</code><br><code>Indeterminate <- {Rule[a, -2], Rule[s, 0.5]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/25.11.E4 25.11.E4] || [[Item:Q7678|<math>\Hurwitzzeta@{s}{a} = \Hurwitzzeta@{s}{a+m}+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{s}}</math>]] || <code>Zeta(0, s, a) = Zeta(0, s, a + m)+ sum((1)/((n + a)^(s)), n = 0..m - 1)</code> || <code>HurwitzZeta[s, a] == HurwitzZeta[s, a + m]+ Sum[Divide[1,(n + a)^(s)], {n, 0, m - 1}, GenerateConditions->None]</code> || Failure || Successful || Error || Successful [Tested: 36]
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| |
| | [https://dlmf.nist.gov/25.11.E5 25.11.E5] || [[Item:Q7679|<math>\Hurwitzzeta@{s}{a} = \sum_{n=0}^{N}\frac{1}{(n+a)^{s}}+\frac{(N+a)^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\floor{x}}{(x+a)^{s+1}}\diff{x}</math>]] || <code>Zeta(0, s, a) = sum((1)/((n + a)^(s)), n = 0..N)+((N + a)^(1 - s))/(s - 1)- s*int((x - floor(x))/((x + a)^(s + 1)), x = N..infinity)</code> || <code>HurwitzZeta[s, a] == Sum[Divide[1,(n + a)^(s)], {n, 0, N}, GenerateConditions->None]+Divide[(N + a)^(1 - s),s - 1]- s*Integrate[Divide[x - Floor[x],(x + a)^(s + 1)], {x, N, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [8 / 9]<div class="mw-collapsible-content"><code>8/9]: [[.2257548023 <- {a = 3/2, s = 3/2, N = 3}</code><br><code>Float(infinity) <- {a = 3/2, s = 1/2, N = 3}</code><br></div></div> || Skipped - Because timed out
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| |-
| |
| | [https://dlmf.nist.gov/25.11.E8 25.11.E8] || [[Item:Q7682|<math>\Hurwitzzeta@{s}{\tfrac{1}{2}a} = \Hurwitzzeta@{s}{\tfrac{1}{2}a+\tfrac{1}{2}}+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}</math>]] || <code>Zeta(0, s, (1)/(2)*a) = Zeta(0, s, (1)/(2)*a +(1)/(2))+ (2)^(s)* sum(((- 1)^(n))/((n + a)^(s)), n = 0..infinity)</code> || <code>HurwitzZeta[s, Divide[1,2]*a] == HurwitzZeta[s, Divide[1,2]*a +Divide[1,2]]+ (2)^(s)* Sum[Divide[(- 1)^(n),(n + a)^(s)], {n, 0, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 3] || Successful [Tested: 3]
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| |-
| |
| | [https://dlmf.nist.gov/25.11.E9 25.11.E9] || [[Item:Q7683|<math>\Hurwitzzeta@{1-s}{a} = \frac{2\EulerGamma@{s}}{(2\pi)^{s}}\*\sum_{n=1}^{\infty}\frac{1}{n^{s}}\cos@{\tfrac{1}{2}\pi s-2n\pi a}</math>]] || <code>Zeta(0, 1 - s, a) = (2*GAMMA(s))/((2*Pi)^(s))* sum((1)/((n)^(s))*cos((1)/(2)*Pi*s - 2*n*Pi*a), n = 1..infinity)</code> || <code>HurwitzZeta[1 - s, a] == Divide[2*Gamma[s],(2*Pi)^(s)]* Sum[Divide[1,(n)^(s)]*Cos[Divide[1,2]*Pi*s - 2*n*Pi*a], {n, 1, Infinity}, GenerateConditions->None]</code> || Error || Failure || - || Skip - No test values generated
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| |-
| |
| | [https://dlmf.nist.gov/25.11.E10 25.11.E10] || [[Item:Q7684|<math>\Hurwitzzeta@{s}{a} = \sum_{n=0}^{\infty}\frac{\Pochhammersym{s}{n}}{n!}\Riemannzeta@{n+s}(1-a)^{n}</math>]] || <code>Zeta(0, s, a) = sum((pochhammer(s, n))/(factorial(n))*Zeta(n + s)*(1 - a)^(n), n = 0..infinity)</code> || <code>HurwitzZeta[s, a] == Sum[Divide[Pochhammer[s, n],(n)!]*Zeta[n + s]*(1 - a)^(n), {n, 0, Infinity}, GenerateConditions->None]</code> || Error || Aborted || - || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 2]<div class="mw-collapsible-content"><code>{Plus[2.612375348685488, Times[-1.0, NSum[Times[Power[0, n], Power[Factorial[n], -1], Pochhammer[1.5, n], Zeta[Plus[1.5, n]]] <- {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, 1], Rule[s, 1.5]}</code><br><code>Plus[1.6449340668482262, Times[-1.0, NSum[Times[Power[0, n], Power[Factorial[n], -1], Pochhammer[2, n], Zeta[Plus[2, n]]] <- {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, 1], Rule[s, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.11.E11 25.11.E11] || [[Item:Q7685|<math>\Hurwitzzeta@{s}{\tfrac{1}{2}} = (2^{s}-1)\Riemannzeta@{s}</math>]] || <code>Zeta(0, s, (1)/(2)) = ((2)^(s)- 1)* Zeta(s)</code> || <code>HurwitzZeta[s, Divide[1,2]] == ((2)^(s)- 1)* Zeta[s]</code> || Successful || Successful || - || Successful [Tested: 6]
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| |-
| |
| | [https://dlmf.nist.gov/25.11.E12 25.11.E12] || [[Item:Q7686|<math>\Hurwitzzeta@{n+1}{a} = \frac{(-1)^{n+1}\digamma^{(n)}@{a}}{n!}</math>]] || <code>Zeta(0, n + 1, a) = ((- 1)^(n + 1)* diff( Psi(a), a$(n) ))/(factorial(n))</code> || <code>HurwitzZeta[n + 1, a] == Divide[(- 1)^(n + 1)* D[PolyGamma[a], {a, n}],(n)!]</code> || Failure || Failure || Error || Successful [Tested: 1]
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| |-
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| | [https://dlmf.nist.gov/25.11.E13 25.11.E13] || [[Item:Q7687|<math>\Hurwitzzeta@{0}{a} = \tfrac{1}{2}-a</math>]] || <code>Zeta(0, 0, a) = (1)/(2)- a</code> || <code>HurwitzZeta[0, a] == Divide[1,2]- a</code> || Successful || Successful || - || Successful [Tested: 6]
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| |-
| |
| | [https://dlmf.nist.gov/25.11.E14 25.11.E14] || [[Item:Q7688|<math>\Hurwitzzeta@{-n}{a} = -\frac{\BernoullipolyB{n+1}@{a}}{n+1}</math>]] || <code>Zeta(0, - n, a) = -(bernoulli(n + 1, a))/(n + 1)</code> || <code>HurwitzZeta[- n, a] == -Divide[BernoulliB[n + 1, a],n + 1]</code> || Successful || Failure || - || Successful [Tested: 1]
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| |-
| |
| | [https://dlmf.nist.gov/25.11.E15 25.11.E15] || [[Item:Q7689|<math>\Hurwitzzeta@{s}{ka} = k^{-s}\*\sum_{n=0}^{k-1}\Hurwitzzeta@{s}{a+\frac{n}{k}}</math>]] || <code>Zeta(0, s, k*a) = (k)^(- s)* sum(Zeta(0, s, a +(n)/(k)), n = 0..k - 1)</code> || <code>HurwitzZeta[s, k*a] == (k)^(- s)* Sum[HurwitzZeta[s, a +Divide[n,k]], {n, 0, k - 1}, GenerateConditions->None]</code> || Failure || Failure || Error || <div class="toccolours mw-collapsible mw-collapsed">Failed [2 / 2]<div class="mw-collapsible-content"><code>{1.3535533905932735 <- {Rule[a, 1], Rule[k, 3], Rule[Times[a, k], 1], Rule[s, 1.5]}</code><br><code>1.2499999999999998 <- {Rule[a, 1], Rule[k, 3], Rule[Times[a, k], 1], Rule[s, 2]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.11.E16 25.11.E16] || [[Item:Q7690|<math>\Hurwitzzeta@{1-s}{\frac{h}{k}} = \frac{2\EulerGamma@{s}}{(2\pi k)^{s}}\*\sum_{r=1}^{k}\cos@{\frac{\pi s}{2}-\frac{2\pi rh}{k}}\Hurwitzzeta@{s}{\frac{r}{k}}</math>]] || <code>Zeta(0, 1 - s, (h)/(k)) = (2*GAMMA(s))/((2*Pi*k)^(s))* sum(cos((Pi*s)/(2)-(2*Pi*r*h)/(k))*Zeta(0, s, (r)/(k)), r = 1..k)</code> || <code>HurwitzZeta[1 - s, Divide[h,k]] == Divide[2*Gamma[s],(2*Pi*k)^(s)]* Sum[Cos[Divide[Pi*s,2]-Divide[2*Pi*r*h,k]]*HurwitzZeta[s, Divide[r,k]], {r, 1, k}, GenerateConditions->None]</code> || Error || Aborted || - || Skipped - Because timed out
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| |
| | [https://dlmf.nist.gov/25.11.E17 25.11.E17] || [[Item:Q7691|<math>\pderiv{}{a}\Hurwitzzeta@{s}{a} = -s\Hurwitzzeta@{s+1}{a}</math>]] || <code>diff(Zeta(0, s, a), a) = - s*Zeta(0, s + 1, a)</code> || <code>D[HurwitzZeta[s, a], a] == - s*HurwitzZeta[s + 1, a]</code> || Error || Successful || - || Successful [Tested: 3]
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| |
| | [https://dlmf.nist.gov/25.11.E18 25.11.E18] || [[Item:Q7692|<math>\Hurwitzzeta'@{0}{a} = \ln@@{\EulerGamma@{a}}-\tfrac{1}{2}\ln@{2\pi}</math>]] || <code>subs( temp=0, diff( Zeta(0, temp, a), temp$(1) ) ) = ln(GAMMA(a))-(1)/(2)*ln(2*Pi)</code> || <code>(D[HurwitzZeta[temp, a], {temp, 1}]/.temp-> 0) == Log[Gamma[a]]-Divide[1,2]*Log[2*Pi]</code> || Failure || Failure || Successful [Tested: 3] || Successful [Tested: 1]
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| | [https://dlmf.nist.gov/25.11.E21 25.11.E21] || [[Item:Q7695|<math>\Hurwitzzeta'@{1-2n}{\frac{h}{k}} = \frac{(\digamma@{2n}-\ln@{2\pi k})\BernoullipolyB{2n}@{h/k}}{2n}-\frac{(\digamma@{2n}-\ln@{2\pi})\BernoullinumberB{2n}}{2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\sin@{\frac{2\pi rh}{k}}\digamma^{(2n-1)}@{\frac{r}{k}}+\frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\cos@{\frac{2\pi rh}{k}}\Hurwitzzeta'@{2n}{\frac{r}{k}}+\frac{\Riemannzeta'@{1-2n}}{k^{2n}}</math>]] || <code>subs( temp=1 - 2*n, diff( Zeta(0, temp, (h)/(k)), temp$(1) ) ) = ((Psi(2*n)- ln(2*Pi*k))* bernoulli(2*n, h/ k))/(2*n)-((Psi(2*n)- ln(2*Pi))* bernoulli(2*n))/(2*n*(k)^(2*n))+((- 1)^(n + 1)* Pi)/((2*Pi*k)^(2*n))*sum(sin((2*Pi*r*h)/(k))*subs( temp=(r)/(k), diff( Psi(temp), temp$(2*n - 1) ) ), r = 1..k - 1)+((- 1)^(n + 1)* 2 *factorial(2*n - 1))/((2*Pi*k)^(2*n))*sum(cos((2*Pi*r*h)/(k))*subs( temp=2*n, diff( Zeta(0, temp, (r)/(k)), temp$(1) ) ), r = 1..k - 1)+(subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) ))/((k)^(2*n))</code> || <code>(D[HurwitzZeta[temp, Divide[h,k]], {temp, 1}]/.temp-> 1 - 2*n) == Divide[(PolyGamma[2*n]- Log[2*Pi*k])* BernoulliB[2*n, h/ k],2*n]-Divide[(PolyGamma[2*n]- Log[2*Pi])* BernoulliB[2*n],2*n*(k)^(2*n)]+Divide[(- 1)^(n + 1)* Pi,(2*Pi*k)^(2*n)]*Sum[Sin[Divide[2*Pi*r*h,k]]*(D[PolyGamma[temp], {temp, 2*n - 1}]/.temp-> Divide[r,k]), {r, 1, k - 1}, GenerateConditions->None]+Divide[(- 1)^(n + 1)* 2 *(2*n - 1)!,(2*Pi*k)^(2*n)]*Sum[Cos[Divide[2*Pi*r*h,k]]*(D[HurwitzZeta[temp, Divide[r,k]], {temp, 1}]/.temp-> 2*n), {r, 1, k - 1}, GenerateConditions->None]+Divide[D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n,(k)^(2*n)]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [70 / 90]<div class="mw-collapsible-content"><code>70/90]: [[-.2303130415-.107731247e-1*I <- {h = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}</code><br><code>.8722916351e-1-.251419603e-1*I <- {h = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/25.11.E22 25.11.E22] || [[Item:Q7696|<math>\Hurwitzzeta'@{1-2n}{\tfrac{1}{2}} = -\frac{\BernoullinumberB{2n}\ln@@{2}}{n\cdot 4^{n}}-\frac{(2^{2n-1}-1)\Riemannzeta'@{1-2n}}{2^{2n-1}}</math>]] || <code>subs( temp=1 - 2*n, diff( Zeta(0, temp, (1)/(2)), temp$(1) ) ) = -(bernoulli(2*n)*ln(2))/(n * (4)^(n))-(((2)^(2*n - 1)- 1)* subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) ))/((2)^(2*n - 1))</code> || <code>(D[HurwitzZeta[temp, Divide[1,2]], {temp, 1}]/.temp-> 1 - 2*n) == -Divide[BernoulliB[2*n]*Log[2],n * (4)^(n)]-Divide[((2)^(2*n - 1)- 1)* (D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n),(2)^(2*n - 1)]</code> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
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| | [https://dlmf.nist.gov/25.11.E23 25.11.E23] || [[Item:Q7697|<math>\Hurwitzzeta'@{1-2n}{\tfrac{1}{3}} = -\frac{\pi(9^{n}-1)\BernoullinumberB{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{\BernoullinumberB{2n}\ln@@{3}}{4n\cdot 3^{2n-1}}-\frac{(-1)^{n}\digamma^{(2n-1)}@{\frac{1}{3}}}{2\sqrt{3}(6\pi)^{2n-1}}-\frac{\left(3^{2n-1}-1\right)\Riemannzeta'@{1-2n}}{2\cdot 3^{2n-1}}</math>]] || <code>subs( temp=1 - 2*n, diff( Zeta(0, temp, (1)/(3)), temp$(1) ) ) = -(Pi*((9)^(n)- 1)* bernoulli(2*n))/(8*n*sqrt(3)*((3)^(2*n - 1)- 1))-(bernoulli(2*n)*ln(3))/(4*n * (3)^(2*n - 1))-((- 1)^(n)* subs( temp=(1)/(3), diff( Psi(temp), temp$(2*n - 1) ) ))/(2*sqrt(3)*(6*Pi)^(2*n - 1))-(((3)^(2*n - 1)- 1)* subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) ))/(2 * (3)^(2*n - 1))</code> || <code>(D[HurwitzZeta[temp, Divide[1,3]], {temp, 1}]/.temp-> 1 - 2*n) == -Divide[Pi*((9)^(n)- 1)* BernoulliB[2*n],8*n*Sqrt[3]*((3)^(2*n - 1)- 1)]-Divide[BernoulliB[2*n]*Log[3],4*n * (3)^(2*n - 1)]-Divide[(- 1)^(n)* (D[PolyGamma[temp], {temp, 2*n - 1}]/.temp-> Divide[1,3]),2*Sqrt[3]*(6*Pi)^(2*n - 1)]-Divide[((3)^(2*n - 1)- 1)* (D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n),2 * (3)^(2*n - 1)]</code> || Failure || Failure || Successful [Tested: 1] || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><code>{Plus[0.010637344739107386, Times[-1.2131199967624389*^-7, D[-3.1320337800208065 <- {0.3333333333333333, 5.0}]]], {Rule[a, 1], Rule[n, 3]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.11.E24 25.11.E24] || [[Item:Q7698|<math>\sum_{r=1}^{k-1}\Hurwitzzeta'@{s}{\frac{r}{k}} = (k^{s}-1)\Riemannzeta'@{s}+k^{s}\Riemannzeta@{s}\ln@@{k}</math>]] || <code>sum(diff( Zeta(0, s, (r)/(k)), s$(1) ), r = 1..k - 1) = ((k)^(s)- 1)* diff( Zeta(s), s$(1) )+ (k)^(s)* Zeta(s)*ln(k)</code> || <code>Sum[D[HurwitzZeta[s, Divide[r,k]], {s, 1}], {r, 1, k - 1}, GenerateConditions->None] == ((k)^(s)- 1)* D[Zeta[s], {s, 1}]+ (k)^(s)* Zeta[s]*Log[k]</code> || Failure || Failure || Successful [Tested: 6] || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><code>{Indeterminate <- {Rule[a, 1], Rule[k, 3], Rule[s, 1]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.11.E25 25.11.E25] || [[Item:Q7699|<math>\Hurwitzzeta@{s}{a} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-e^{-x}}\diff{x}</math>]] || <code>Zeta(0, s, a) = (1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 - exp(- x)), x = 0..infinity)</code> || <code>HurwitzZeta[s, a] == Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 - Exp[- x]], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Successful [Tested: 6] || Successful [Tested: 6]
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| | [https://dlmf.nist.gov/25.11.E26 25.11.E26] || [[Item:Q7700|<math>\Hurwitzzeta@{s}{a} = -s\int_{-a}^{\infty}\frac{x-\floor{x}-\frac{1}{2}}{(x+a)^{s+1}}\diff{x}</math>]] || <code>Zeta(0, s, a) = - s*int((x - floor(x)-(1)/(2))/((x + a)^(s + 1)), x = - a..infinity)</code> || <code>HurwitzZeta[s, a] == - s*Integrate[Divide[x - Floor[x]-Divide[1,2],(x + a)^(s + 1)], {x, - a, Infinity}, GenerateConditions->None]</code> || Error || Aborted || - || Skip - No test values generated
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| | [https://dlmf.nist.gov/25.11.E27 25.11.E27] || [[Item:Q7701|<math>\Hurwitzzeta@{s}{a} = \frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{ax}}\diff{x}</math>]] || <code>Zeta(0, s, a) = (1)/(2)*(a)^(- s)+((a)^(1 - s))/(s - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2))*((x)^(s - 1))/(exp(a*x)), x = 0..infinity)</code> || <code>HurwitzZeta[s, a] == Divide[1,2]*(a)^(- s)+Divide[(a)^(1 - s),s - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2])*Divide[(x)^(s - 1),Exp[a*x]], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/25.11.E28 25.11.E28] || [[Item:Q7702|<math>\Hurwitzzeta@{s}{a} = \frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\sum_{k=1}^{n}\frac{\BernoullinumberB{2k}}{(2k)!}\Pochhammersym{s}{2k-1}a^{1-s-2k}+\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{k=1}^{n}\frac{\BernoullinumberB{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}e^{-ax}\diff{x}</math>]] || <code>Zeta(0, s, a) = (1)/(2)*(a)^(- s)+((a)^(1 - s))/(s - 1)+ sum((bernoulli(2*k))/(factorial(2*k))*pochhammer(s, 2*k - 1)*(a)^(1 - s - 2*k)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2)- sum((bernoulli(2*k))/(factorial(2*k))*(x)^(2*k - 1), k = 1..n))* (x)^(s - 1)* exp(- a*x), x = 0..infinity), k = 1..n)</code> || <code>HurwitzZeta[s, a] == Divide[1,2]*(a)^(- s)+Divide[(a)^(1 - s),s - 1]+ Sum[Divide[BernoulliB[2*k],(2*k)!]*Pochhammer[s, 2*k - 1]*(a)^(1 - s - 2*k)+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2]- Sum[Divide[BernoulliB[2*k],(2*k)!]*(x)^(2*k - 1), {k, 1, n}, GenerateConditions->None])* (x)^(s - 1)* Exp[- a*x], {x, 0, Infinity}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]</code> || Error || Aborted || - || Skipped - Because timed out
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| | [https://dlmf.nist.gov/25.11.E29 25.11.E29] || [[Item:Q7703|<math>\Hurwitzzeta@{s}{a} = \frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+2\int_{0}^{\infty}\frac{\sin@{s\atan@{x/a}}}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\diff{x}</math>]] || <code>Zeta(0, s, a) = (1)/(2)*(a)^(- s)+((a)^(1 - s))/(s - 1)+ 2*int((sin(s*arctan(x/ a)))/(((a)^(2)+ (x)^(2))^(s/ 2)*(exp(2*Pi*x)- 1)), x = 0..infinity)</code> || <code>HurwitzZeta[s, a] == Divide[1,2]*(a)^(- s)+Divide[(a)^(1 - s),s - 1]+ 2*Integrate[Divide[Sin[s*ArcTan[x/ a]],((a)^(2)+ (x)^(2))^(s/ 2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
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| | [https://dlmf.nist.gov/25.11.E30 25.11.E30] || [[Item:Q7704|<math>\Hurwitzzeta@{s}{a} = \frac{\EulerGamma@{1-s}}{2\pi i}\int_{-\infty}^{(0+)}\frac{e^{az}z^{s-1}}{1-e^{z}}\diff{z}</math>]] || <code>Zeta(0, s, a) = (GAMMA(1 - s))/(2*Pi*I)*int((exp(a*z)*(z)^(s - 1))/(1 - exp(z)), z = - infinity..(0 +))</code> || <code>HurwitzZeta[s, a] == Divide[Gamma[1 - s],2*Pi*I]*Integrate[Divide[Exp[a*z]*(z)^(s - 1),1 - Exp[z]], {z, - Infinity, (0 +)}, GenerateConditions->None]</code> || Error || Failure || - || Error
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| | [https://dlmf.nist.gov/25.11.E31 25.11.E31] || [[Item:Q7705|<math>\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{2\cosh@@{x}}\diff{x} = 4^{-s}\left(\Hurwitzzeta@{s}{\tfrac{1}{4}+\tfrac{1}{4}a}-\Hurwitzzeta@{s}{\tfrac{3}{4}+\tfrac{1}{4}a}\right)</math>]] || <code>(1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(2*cosh(x)), x = 0..infinity) = (4)^(- s)*(Zeta(0, s, (1)/(4)+(1)/(4)*a)- Zeta(0, s, (3)/(4)+(1)/(4)*a))</code> || <code>Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],2*Cosh[x]], {x, 0, Infinity}, GenerateConditions->None] == (4)^(- s)*(HurwitzZeta[s, Divide[1,4]+Divide[1,4]*a]- HurwitzZeta[s, Divide[3,4]+Divide[1,4]*a])</code> || Failure || Successful || Skipped - Because timed out || Successful [Tested: 12]
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| | [https://dlmf.nist.gov/25.11.E32 25.11.E32] || [[Item:Q7706|<math>\int_{0}^{a}x^{n}\digamma@{x}\diff{x} = (-1)^{n-1}\Riemannzeta'@{-n}+(-1)^{n}h(n)\frac{\BernoullinumberB{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}h(k)\frac{\BernoullinumberB{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}\binom{n}{k}\Hurwitzzeta'@{-k}{a}a^{n-k}</math>]] || <code>int((x)^(n)* Psi(x), x = 0..a) = (- 1)^(n - 1)* subs( temp=- n, diff( Zeta(temp), temp$(1) ) )+(- 1)^(n)* h*(n)*(bernoulli(n + 1))/(n + 1)- sum((- 1)^(k)*binomial(n,k)*h*(k)*(bernoulli(k + 1)*(a))/(k + 1)*(a)^(n - k), k = 0..n)+ sum((- 1)^(k)*binomial(n,k)*subs( temp=- k, diff( Zeta(0, temp, a), temp$(1) ) )*(a)^(n - k), k = 0..n)</code> || <code>Integrate[(x)^(n)* PolyGamma[x], {x, 0, a}, GenerateConditions->None] == (- 1)^(n - 1)* (D[Zeta[temp], {temp, 1}]/.temp-> - n)+(- 1)^(n)* h*(n)*Divide[BernoulliB[n + 1],n + 1]- Sum[(- 1)^(k)*Binomial[n,k]*h*(k)*Divide[BernoulliB[k + 1]*(a),k + 1]*(a)^(n - k), {k, 0, n}, GenerateConditions->None]+ Sum[(- 1)^(k)*Binomial[n,k]*(D[HurwitzZeta[temp, a], {temp, 1}]/.temp-> - k)*(a)^(n - k), {k, 0, n}, GenerateConditions->None]</code> || Failure || Aborted || <div class="toccolours mw-collapsible mw-collapsed">Failed [30 / 30]<div class="mw-collapsible-content"><code>30/30]: [[.9441788834-.4156250000*I <- {a = 3/2, h = 1/2*3^(1/2)+1/2*I, n = 3}</code><br><code>2.079687501-.7198836171*I <- {a = 3/2, h = -1/2+1/2*I*3^(1/2), n = 3}</code><br></div></div> || Skipped - Because timed out
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| | [https://dlmf.nist.gov/25.11.E33 25.11.E33] || [[Item:Q7707|<math>h(n) = \sum_{k=1}^{n}k^{-1}</math>]] || <code>h*(n) = sum((k)^(- 1), k = 1..n)</code> || <code>h*(n) == Sum[(k)^(- 1), {k, 1, n}, GenerateConditions->None]</code> || Skipped - no semantic math || Skipped - no semantic math || - || -
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| | [https://dlmf.nist.gov/25.11.E34 25.11.E34] || [[Item:Q7708|<math>n\int_{0}^{a}\Hurwitzzeta'@{1-n}{x}\diff{x} = \Hurwitzzeta'@{-n}{a}-\Riemannzeta'@{-n}+\frac{\BernoullinumberB{n+1}-\BernoullipolyB{n+1}@{a}}{n(n+1)}</math>]] || <code>n*int(subs( temp=1 - n, diff( Zeta(0, temp, x), temp$(1) ) ), x = 0..a) = subs( temp=- n, diff( Zeta(0, temp, a), temp$(1) ) )- subs( temp=- n, diff( Zeta(temp), temp$(1) ) )+(bernoulli(n + 1)- bernoulli(n + 1, a))/(n*(n + 1))</code> || <code>n*Integrate[D[HurwitzZeta[temp, x], {temp, 1}]/.temp-> 1 - n, {x, 0, a}, GenerateConditions->None] == (D[HurwitzZeta[temp, a], {temp, 1}]/.temp-> - n)- (D[Zeta[temp], {temp, 1}]/.temp-> - n)+Divide[BernoulliB[n + 1]- BernoulliB[n + 1, a],n*(n + 1)]</code> || Failure || Failure || Manual Skip! || Successful [Tested: 3]
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| | [https://dlmf.nist.gov/25.11.E35 25.11.E35] || [[Item:Q7709|<math>\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\diff{x}</math>]] || <code>sum(((- 1)^(n))/((n + a)^(s)), n = 0..infinity) = (1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 + exp(- x)), x = 0..infinity)</code> || <code>Sum[Divide[(- 1)^(n),(n + a)^(s)], {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 + Exp[- x]], {x, 0, Infinity}, GenerateConditions->None]</code> || Error || Successful || - || -
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| | [https://dlmf.nist.gov/25.11.E35 25.11.E35] || [[Item:Q7709|<math>\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\diff{x} = 2^{-s}\left(\Hurwitzzeta@{s}{\tfrac{1}{2}a}-\Hurwitzzeta@{s}{\tfrac{1}{2}(1+a)}\right)</math>]] || <code>(1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 + exp(- x)), x = 0..infinity) = (2)^(- s)*(Zeta(0, s, (1)/(2)*a)- Zeta(0, s, (1)/(2)*(1 + a)))</code> || <code>Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 + Exp[- x]], {x, 0, Infinity}, GenerateConditions->None] == (2)^(- s)*(HurwitzZeta[s, Divide[1,2]*a]- HurwitzZeta[s, Divide[1,2]*(1 + a)])</code> || Error || Successful || - || -
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| | [https://dlmf.nist.gov/25.11.E36 25.11.E36] || [[Item:Q7710|<math>\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}} = k^{-s}\sum_{r=1}^{k-1}\chi(r)\Hurwitzzeta@{s}{\frac{r}{k}}</math>]] || <code>sum((chi*(n))/((n)^(s)), n = 1..infinity) = (k)^(- s)* sum(chi*(r)* Zeta(0, s, (r)/(k)), r = 1..k - 1)</code> || <code>Sum[Divide[\[Chi]*(n),(n)^(s)], {n, 1, Infinity}, GenerateConditions->None] == (k)^(- s)* Sum[\[Chi]*(r)* HurwitzZeta[s, Divide[r,k]], {r, 1, k - 1}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 60]<div class="mw-collapsible-content"><code>60/60]: [[Float(infinity)+Float(infinity)*I <- {chi = 1/2*3^(1/2)+1/2*I, s = 3/2, k = 1}</code><br><code>Float(infinity)+Float(infinity)*I <- {chi = 1/2*3^(1/2)+1/2*I, s = 3/2, k = 2}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [60 / 60]<div class="mw-collapsible-content"><code>{Complex[-1.264704103160249, -0.7301772544047939] <- {Rule[a, 1], Rule[k, 1], Rule[s, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[-2.727214191729021, -1.574557847732518] <- {Rule[a, 1], Rule[k, 2], Rule[s, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.11.E37 25.11.E37] || [[Item:Q7711|<math>\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\Hurwitzzeta@{nk}{a} = -n\ln@@{\EulerGamma@{a}}+\ln@{\prod_{j=0}^{n-1}\EulerGamma@{a-e^{(2j+1)\pi i/n}}}</math>]] || <code>sum(((- 1)^(k))/(k)*Zeta(0, n*k, a), k = 1..infinity) = - n*ln(GAMMA(a))+ ln(product(GAMMA(a - exp((2*j + 1)* Pi*I/ n)), j = 0..n - 1))</code> || <code>Sum[Divide[(- 1)^(k),k]*HurwitzZeta[n*k, a], {k, 1, Infinity}, GenerateConditions->None] == - n*Log[Gamma[a]]+ Log[Product[Gamma[a - Exp[(2*j + 1)* Pi*I/ n]], {j, 0, n - 1}, GenerateConditions->None]]</code> || Failure || Failure || Successful [Tested: 2] || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 3]<div class="mw-collapsible-content"><code>{NSum[Times[Power[-1, k], Power[k, -1], Zeta[k]] <- {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]], {Rule[a, 1], Rule[n, 1]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.11.E38 25.11.E38] || [[Item:Q7712|<math>\sum_{k=1}^{\infty}\binom{n+k}{k}\Hurwitzzeta@{n+k+1}{a}z^{k} = \frac{(-1)^{n}}{n!}\left(\digamma^{(n)}@{a}-\digamma^{(n)}@{a-z}\right)</math>]] || <code>sum(binomial(n + k,k)*Zeta(0, n + k + 1, a)*(z)^(k), k = 1..infinity) = ((- 1)^(n))/(factorial(n))*(diff( Psi(a), a$(n) )- subs( temp=a - z, diff( Psi(temp), temp$(n) ) ))</code> || <code>Sum[Binomial[n + k,k]*HurwitzZeta[n + k + 1, a]*(z)^(k), {k, 1, Infinity}, GenerateConditions->None] == Divide[(- 1)^(n),(n)!]*(D[PolyGamma[a], {a, n}]- (D[PolyGamma[temp], {temp, n}]/.temp-> a - z))</code> || Failure || Failure || Manual Skip! || Skipped - Because timed out
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| | [https://dlmf.nist.gov/25.11.E39 25.11.E39] || [[Item:Q7713|<math>\sum_{k=2}^{\infty}\frac{k}{2^{k}}\Hurwitzzeta@{k+1}{\tfrac{3}{4}} = 8G</math>]] || <code>sum((k)/((2)^(k))*Zeta(0, k + 1, (3)/(4)), k = 2..infinity) = 8*G</code> || <code>Sum[Divide[k,(2)^(k)]*HurwitzZeta[k + 1, Divide[3,4]], {k, 2, Infinity}, GenerateConditions->None] == 8*G</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>10/10]: [[.399521521-4.000000000*I <- {G = 1/2*3^(1/2)+1/2*I}</code><br><code>11.32772475-6.928203232*I <- {G = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Complex[0.39952152314224243, -3.9999999999999996] <- {Rule[a, 1], Rule[G, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Complex[11.327724753417751, -6.92820323027551] <- {Rule[a, 1], Rule[G, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/25.11.E40 25.11.E40] || [[Item:Q7714|<math>G\defeq\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}} = 0.91596\;55941\;772\dots</math>]] || <code>G = sum(((- 1)^(n))/((2*n + 1)^(2)), n = 0..infinity) = 0.9159655941772</code> || <code>G == Sum[Divide[(- 1)^(n),(2*n + 1)^(2)], {n, 0, Infinity}, GenerateConditions->None] == 0.9159655941772</code> || Failure || Skipped - Invalid test case: dots || Error || -
| |
| |-
| |
| | [https://dlmf.nist.gov/25.12.E2 25.12.E2] || [[Item:Q7721|<math>\dilog@{z} = -\int_{0}^{z}t^{-1}\ln@{1-t}\diff{t}</math>]] || <code>dilog(z) = - int((t)^(- 1)* ln(1 - t), t = 0..z)</code> || <code>PolyLog[2, z] == - Integrate[(t)^(- 1)* Log[1 - t], {t, 0, z}, GenerateConditions->None]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 7]<div class="mw-collapsible-content"><code>6/7]: [[-.8224670339-1.383979491*I <- {z = 1/2*3^(1/2)+1/2*I}</code><br><code>1.644934067-2.503719574*I <- {z = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || Successful [Tested: 7]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.12.E3 25.12.E3] || [[Item:Q7722|<math>\dilog@{z}+\dilog@{\frac{z}{z-1}} = -\frac{1}{2}(\ln@{1-z})^{2}</math>]] || <code>dilog(z)+ dilog((z)/(z - 1)) = -(1)/(2)*(ln(1 - z))^(2)</code> || <code>PolyLog[2, z]+ PolyLog[2, Divide[z,z - 1]] == -Divide[1,2]*(Log[1 - z])^(2)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><code>1/1]: [[3.289868134-2.177586090*I <- {z = 1/2}</code><br></div></div> || Successful [Tested: 1]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.12.E4 25.12.E4] || [[Item:Q7723|<math>\dilog@{z}+\dilog@{\frac{1}{z}} = -\frac{1}{6}\pi^{2}-\frac{1}{2}(\ln@{-z})^{2}</math>]] || <code>dilog(z)+ dilog((1)/(z)) = -(1)/(6)*(Pi)^(2)-(1)/(2)*(ln(- z))^(2)</code> || <code>PolyLog[2, z]+ PolyLog[2, Divide[1,z]] == -Divide[1,6]*(Pi)^(2)-Divide[1,2]*(Log[- z])^(2)</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><code>1/1]: [[6.579736268-4.725198502*I <- {z = -1/2}</code><br></div></div> || Successful [Tested: 1]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.12.E5 25.12.E5] || [[Item:Q7724|<math>\dilog@{z^{m}} = m\sum_{k=0}^{m-1}\dilog@{ze^{2\pi ik/m}}</math>]] || <code>dilog((z)^(m)) = m*sum(dilog(z*exp(2*Pi*I*k/ m)), k = 0..m - 1)</code> || <code>PolyLog[2, (z)^(m)] == m*Sum[PolyLog[2, z*Exp[2*Pi*I*k/ m]], {k, 0, m - 1}, GenerateConditions->None]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [1 / 1]<div class="mw-collapsible-content"><code>1/1]: [[-8.968925063+0.*I <- {z = 1/2, m = 3}</code><br></div></div> || Successful [Tested: 1]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.12.E6 25.12.E6] || [[Item:Q7725|<math>\dilog@{x}+\dilog@{1-x} = \frac{1}{6}\pi^{2}-(\ln@@{x})\ln@{1-x}</math>]] || <code>dilog(x)+ dilog(1 - x) = (1)/(6)*(Pi)^(2)-(ln(x))* ln(1 - x)</code> || <code>PolyLog[2, x]+ PolyLog[2, 1 - x] == Divide[1,6]*(Pi)^(2)-(Log[x])* Log[1 - x]</code> || Successful || Successful || - || Successful [Tested: 1]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.12.E7 25.12.E7] || [[Item:Q7726|<math>\dilog@{e^{i\theta}} = \sum_{n=1}^{\infty}\frac{\cos@{n\theta}}{n^{2}}+i\sum_{n=1}^{\infty}\frac{\sin@{n\theta}}{n^{2}}</math>]] || <code>dilog(exp(I*theta)) = sum((cos(n*theta))/((n)^(2)), n = 1..infinity)+ I*sum((sin(n*theta))/((n)^(2)), n = 1..infinity)</code> || <code>PolyLog[2, Exp[I*\[Theta]]] == Sum[Divide[Cos[n*\[Theta]],(n)^(2)], {n, 1, Infinity}, GenerateConditions->None]+ I*Sum[Divide[Sin[n*\[Theta]],(n)^(2)], {n, 1, Infinity}, GenerateConditions->None]</code> || Failure || Successful || Skipped - Because timed out || Successful [Tested: 10]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.12.E8 25.12.E8] || [[Item:Q7727|<math>\sum_{n=1}^{\infty}\frac{\cos@{n\theta}}{n^{2}} = \frac{\pi^{2}}{6}-\frac{\pi\theta}{2}+\frac{\theta^{2}}{4}</math>]] || <code>sum((cos(n*theta))/((n)^(2)), n = 1..infinity) = ((Pi)^(2))/(6)-(Pi*theta)/(2)+((theta)^(2))/(4)</code> || <code>Sum[Divide[Cos[n*\[Theta]],(n)^(2)], {n, 1, Infinity}, GenerateConditions->None] == Divide[(Pi)^(2),6]-Divide[Pi*\[Theta],2]+Divide[\[Theta]^(2),4]</code> || Failure || Failure || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [5 / 10]<div class="mw-collapsible-content"><code>{Complex[-1.5707963267948957, 2.720699046351327] <- {Rule[θ, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br><code>Complex[-2.720699046351327, -1.5707963267948966] <- {Rule[θ, Power[E, Times[Complex[0, Rational[-5, 6]], Pi]]]}</code><br></div></div>
| |
| |-
| |
| | [https://dlmf.nist.gov/25.12.E9 25.12.E9] || [[Item:Q7728|<math>\sum_{n=1}^{\infty}\frac{\sin@{n\theta}}{n^{2}} = -\int_{0}^{\theta}\ln@{2\sin@{\tfrac{1}{2}x}}\diff{x}</math>]] || <code>sum((sin(n*theta))/((n)^(2)), n = 1..infinity) = - int(ln(2*sin((1)/(2)*x)), x = 0..theta)</code> || <code>Sum[Divide[Sin[n*\[Theta]],(n)^(2)], {n, 1, Infinity}, GenerateConditions->None] == - Integrate[Log[2*Sin[Divide[1,2]*x]], {x, 0, \[Theta]}, GenerateConditions->None]</code> || Failure || Aborted || Skipped - Because timed out || Skipped - Because timed out
| |
| |-
| |
| | [https://dlmf.nist.gov/25.12.E10 25.12.E10] || [[Item:Q7729|<math>\polylog{s}@{z} = \sum_{n=1}^{\infty}\frac{z^{n}}{n^{s}}</math>]] || <code>polylog(s, z) = sum(((z)^(n))/((n)^(s)), n = 1..infinity)</code> || <code>PolyLog[s, z] == Sum[Divide[(z)^(n),(n)^(s)], {n, 1, Infinity}, GenerateConditions->None]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [12 / 42]<div class="mw-collapsible-content"><code>12/42]: [[Float(-infinity)-12.69850170*I <- {s = -3/2, z = 3/2}</code><br><code>Float(-infinity)-3.323322953*I <- {s = -3/2, z = 2}</code><br></div></div> || Successful [Tested: 42]
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| |-
| |
| | [https://dlmf.nist.gov/25.12.E12 25.12.E12] || [[Item:Q7731|<math>\polylog{s}@{z} = \EulerGamma@{1-s}\left(\ln@@{\frac{1}{z}}\right)^{s-1}+\sum_{n=0}^{\infty}\Riemannzeta@{s-n}\frac{(\ln@@{z})^{n}}{n!}</math>]] || <code>polylog(s, z) = GAMMA(1 - s)*(ln((1)/(z)))^(s - 1)+ sum(Zeta(s - n)*((ln(z))^(n))/(factorial(n)), n = 0..infinity)</code> || <code>PolyLog[s, z] == Gamma[1 - s]*(Log[Divide[1,z]])^(s - 1)+ Sum[Zeta[s - n]*Divide[(Log[z])^(n),(n)!], {n, 0, Infinity}, GenerateConditions->None]</code> || Failure || Failure || Successful [Tested: 7] || Skipped - Because timed out
| |
| |-
| |
| | [https://dlmf.nist.gov/25.12.E13 25.12.E13] || [[Item:Q7732|<math>\polylog{s}@{e^{2\pi ia}}+e^{\pi is}\polylog{s}@{e^{-2\pi ia}} = \frac{(2\pi)^{s}e^{\pi is/2}}{\EulerGamma@{s}}\Hurwitzzeta@{1-s}{a}</math>]] || <code>polylog(s, exp(2*Pi*I*a))+ exp(Pi*I*s)*polylog(s, exp(- 2*Pi*I*a)) = ((2*Pi)^(s)* exp(Pi*I*s/ 2))/(GAMMA(s))*Zeta(0, 1 - s, a)</code> || <code>PolyLog[s, Exp[2*Pi*I*a]]+ Exp[Pi*I*s]*PolyLog[s, Exp[- 2*Pi*I*a]] == Divide[(2*Pi)^(s)* Exp[Pi*I*s/ 2],Gamma[s]]*HurwitzZeta[1 - s, a]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [15 / 18]<div class="mw-collapsible-content"><code>15/18]: [[24.27636385+24.27636386*I <- {a = -3/2, s = 3/2}</code><br><code>-2.230710143+2.230710142*I <- {a = -3/2, s = 1/2}</code><br></div></div> || Skip - No test values generated
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| |-
| |
| | [https://dlmf.nist.gov/25.12#Ex1 25.12#Ex1] || [[Item:Q7735|<math>F_{s}(x) = -\polylog{s+1}@{-e^{x}}</math>]] || <code>F[s]*(x) = - polylog(s + 1, - exp(x))</code> || <code>Subscript[F, s]*(x) == - PolyLog[s + 1, - Exp[x]]</code> || Failure || Successful || Error || Successful [Tested: 0]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.12#Ex2 25.12#Ex2] || [[Item:Q7736|<math>G_{s}(x) = \polylog{s+1}@{e^{x}}</math>]] || <code>G[s]*(x) = polylog(s + 1, exp(x))</code> || <code>Subscript[G, s]*(x) == PolyLog[s + 1, Exp[x]]</code> || Failure || Successful || Error || Successful [Tested: 0]
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| |-
| |
| | [https://dlmf.nist.gov/25.14.E2 25.14.E2] || [[Item:Q7741|<math>\Hurwitzzeta@{s}{a} = \LerchPhi@{1}{s}{a}</math>]] || <code>Zeta(0, s, a) = LerchPhi(1, s, a)</code> || <code>HurwitzZeta[s, a] == LerchPhi[1, s, a]</code> || Successful || Failure || - || Successful [Tested: 2]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.14.E3 25.14.E3] || [[Item:Q7742|<math>\polylog{s}@{z} = z\LerchPhi@{z}{s}{1}</math>]] || <code>polylog(s, z) = z*LerchPhi(z, s, 1)</code> || <code>PolyLog[s, z] == z*LerchPhi[z, s, 1]</code> || Successful || Successful || - || Successful [Tested: 10]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.14.E4 25.14.E4] || [[Item:Q7743|<math>\LerchPhi@{z}{s}{a} = z^{m}\LerchPhi@{z}{s}{a+m}+\sum_{n=0}^{m-1}\frac{z^{n}}{(a+n)^{s}}</math>]] || <code>LerchPhi(z, s, a) = (z)^(m)* LerchPhi(z, s, a + m)+ sum(((z)^(n))/((a + n)^(s)), n = 0..m - 1)</code> || <code>LerchPhi[z, s, a] == (z)^(m)* LerchPhi[z, s, a + m]+ Sum[Divide[(z)^(n),(a + n)^(s)], {n, 0, m - 1}, GenerateConditions->None]</code> || Failure || Successful || <div class="toccolours mw-collapsible mw-collapsed">Failed [6 / 300]<div class="mw-collapsible-content"><code>6/300]: [[.27656730e-2-.27656730e-2*I <- {a = -3/2, s = -2, z = 1/2*3^(1/2)+1/2*I, m = 2}</code><br><code>0.+.228647547e-1*I <- {a = -3/2, s = -2, z = -1/2+1/2*I*3^(1/2), m = 2}</code><br></div></div> || Successful [Tested: 300]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.14.E5 25.14.E5] || [[Item:Q7744|<math>\LerchPhi@{z}{s}{a} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-ze^{-x}}\diff{x}</math>]] || <code>LerchPhi(x + y*I, s, a) = (1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 -(x + y*I)* exp(- x)), x = 0..infinity)</code> || <code>LerchPhi[x + y*I, s, a] == Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 -(x + y*I)* Exp[- x]], {x, 0, Infinity}, GenerateConditions->None]</code> || Failure || Aborted || Skipped - Because timed out || <div class="toccolours mw-collapsible mw-collapsed">Failed [162 / 162]<div class="mw-collapsible-content"><code>{Plus[Complex[0.29818646299224294, -0.45270555517796296], Times[-1.1283791670955126, NIntegrate[Complex[0.15484016278663867, -0.07789552790412994] <- {1.5, 0, DirectedInfinity[1]}]]], {Rule[a, 1.5], Rule[s, 1.5], Rule[x, 1.5], Rule[y, -1.5], Rule[z, Rational[1, 2]]}</code><br><code>Plus[Complex[0.29818646299224244, 0.45270555517796246], Times[-1.1283791670955126, NIntegrate[Complex[0.15484016278663867, 0.07789552790412994] <- {1.5, 0, DirectedInfinity[1]}]]], {Rule[a, 1.5], Rule[s, 1.5], Rule[x, 1.5], Rule[y, 1.5], Rule[z, Rational[1, 2]]}</code><br></div></div>
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| |-
| |
| | [https://dlmf.nist.gov/25.14.E6 25.14.E6] || [[Item:Q7745|<math>\LerchPhi@{z}{s}{a} = \frac{1}{2}a^{-s}+\int_{0}^{\infty}\frac{z^{x}}{(a+x)^{s}}\diff{x}-2\int_{0}^{\infty}\frac{\sin@{x\ln@@{z}-s\atan@{x/a}}}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\diff{x}</math>]] || <code>LerchPhi(x + y*I, s, a) = (1)/(2)*(a)^(- s)+ int(((x + y*I)^(x))/((a + x)^(s)), x = 0..infinity)- 2*int((sin(x*ln(x + y*I)- s*arctan(x/ a)))/(((a)^(2)+ (x)^(2))^(s/ 2)*(exp(2*Pi*x)- 1)), x = 0..infinity)</code> || <code>LerchPhi[x + y*I, s, a] == Divide[1,2]*(a)^(- s)+ Integrate[Divide[(x + y*I)^(x),(a + x)^(s)], {x, 0, Infinity}, GenerateConditions->None]- 2*Integrate[Divide[Sin[x*Log[x + y*I]- s*ArcTan[x/ a]],((a)^(2)+ (x)^(2))^(s/ 2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}, GenerateConditions->None]</code> || Error || Aborted || - || Skipped - Because timed out
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| |-
| |
| | [https://dlmf.nist.gov/25.16.E10 25.16.E10] || [[Item:Q7766|<math>\frac{1}{2}\Riemannzeta@{1-2a} = -\frac{\BernoullinumberB{2a}}{4a}</math>]] || <code>(1)/(2)*Zeta(1 - 2*a) = -(bernoulli(2*a))/(4*a)</code> || <code>Divide[1,2]*Zeta[1 - 2*a] == -Divide[BernoulliB[2*a],4*a]</code> || Failure || Failure || Successful [Tested: 1] || Successful [Tested: 1]
| |
| |-
| |
| | [https://dlmf.nist.gov/25.16.E13 25.16.E13] || [[Item:Q7769|<math>\sum_{n=1}^{\infty}\left(\frac{h(n)}{n}\right)^{2} = \frac{17}{4}\Riemannzeta@{4}</math>]] || <code>sum(((h*(n))/(n))^(2), n = 1..infinity) = (17)/(4)*Zeta(4)</code> || <code>Sum[(Divide[h*(n),n])^(2), {n, 1, Infinity}, GenerateConditions->None] == Divide[17,4]*Zeta[4]</code> || Failure || Failure || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>10/10]: [[Float(infinity)+Float(infinity)*I <- {h = 1/2*3^(1/2)+1/2*I}</code><br><code>Float(infinity)+Float(infinity)*I <- {h = -1/2+1/2*I*3^(1/2)}</code><br></div></div> || <div class="toccolours mw-collapsible mw-collapsed">Failed [10 / 10]<div class="mw-collapsible-content"><code>{Plus[-4.599873743272337, NSum[Power[E, Times[Complex[0, Rational[1, 3]], Pi]] <- {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[h, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}</code><br><code>Plus[-4.599873743272337, NSum[Power[E, Times[Complex[0, Rational[-2, 3]], Pi]] <- {n, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]]], {Rule[h, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}</code><br></div></div>
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| | [https://dlmf.nist.gov/25.16.E14 25.16.E14] || [[Item:Q7770|<math>\sum_{r=1}^{\infty}\sum_{k=1}^{r}\frac{1}{rk(r+k)} = \frac{5}{4}\Riemannzeta@{3}</math>]] || <code>sum(sum((1)/(r*k*(r + k)), k = 1..r), r = 1..infinity) = (5)/(4)*Zeta(3)</code> || <code>Sum[Sum[Divide[1,r*k*(r + k)], {k, 1, r}, GenerateConditions->None], {r, 1, Infinity}, GenerateConditions->None] == Divide[5,4]*Zeta[3]</code> || Failure || Aborted || Error || Successful [Tested: 1] | |
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| |}
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