Zeta and Related Functions - 25.5 Integral Representations

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25.5.E1 ζ ⁡ ( s ) = 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 e x - 1 ⁢ d x Riemann-zeta 𝑠 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 superscript 𝑒 𝑥 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{\Gamma\left(s\right)}% \int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}\mathrm{d}x}}
\Riemannzeta@{s} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}\diff{x}		 ℜ ⁡ s > 1 , ℜ ⁡ s > 0 formulae-sequence 𝑠 1 𝑠 0 {\displaystyle{\displaystyle\Re s>1,\Re s>0}} 
Zeta(s) = (1)/(GAMMA(s))*int(((x)^(s - 1))/(exp(x)- 1), x = 0..infinity)

Zeta[s] == Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]- 1], {x, 0, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 2] Successful [Tested: 2]
25.5.E2 ζ ⁡ ( s ) = 1 Γ ⁡ ( s + 1 ) ⁢ ∫ 0 ∞ e x ⁢ x s ( e x - 1 ) 2 ⁢ d x Riemann-zeta 𝑠 1 Euler-Gamma 𝑠 1 superscript subscript 0 superscript 𝑒 𝑥 superscript 𝑥 𝑠 superscript superscript 𝑒 𝑥 1 2 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{\Gamma\left(s+1\right% )}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}-1)^{2}}\mathrm{d}x}}
\Riemannzeta@{s} = \frac{1}{\EulerGamma@{s+1}}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}-1)^{2}}\diff{x}		 ℜ ⁡ s > 1 , ℜ ⁡ ( s + 1 ) > 0 formulae-sequence 𝑠 1 𝑠 1 0 {\displaystyle{\displaystyle\Re s>1,\Re(s+1)>0}} 
Zeta(s) = (1)/(GAMMA(s + 1))*int((exp(x)*(x)^(s))/((exp(x)- 1)^(2)), x = 0..infinity)

Zeta[s] == Divide[1,Gamma[s + 1]]*Integrate[Divide[Exp[x]*(x)^(s),(Exp[x]- 1)^(2)], {x, 0, Infinity}, GenerateConditions->None]
Failure Aborted Successful [Tested: 2] Successful [Tested: 2]
25.5.E3 ζ ⁡ ( s ) = 1 ( 1 - 2 1 - s ) ⁢ Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 e x + 1 ⁢ d x Riemann-zeta 𝑠 1 1 superscript 2 1 𝑠 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 superscript 𝑒 𝑥 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma% \left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}+1}\mathrm{d}x}}
\Riemannzeta@{s} = \frac{1}{(1-2^{1-s})\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}+1}\diff{x}		 ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle\Re s>0}} 
Zeta(s) = (1)/((1 - (2)^(1 - s))*GAMMA(s))*int(((x)^(s - 1))/(exp(x)+ 1), x = 0..infinity)

Zeta[s] == Divide[1,(1 - (2)^(1 - s))*Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]+ 1], {x, 0, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
25.5.E4 ζ ⁡ ( s ) = 1 ( 1 - 2 1 - s ) ⁢ Γ ⁡ ( s + 1 ) ⁢ ∫ 0 ∞ e x ⁢ x s ( e x + 1 ) 2 ⁢ d x Riemann-zeta 𝑠 1 1 superscript 2 1 𝑠 Euler-Gamma 𝑠 1 superscript subscript 0 superscript 𝑒 𝑥 superscript 𝑥 𝑠 superscript superscript 𝑒 𝑥 1 2 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma% \left(s+1\right)}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}+1)^{2}}\mathrm{d}x}}
\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}		 ℜ ⁡ s > 0 , ℜ ⁡ ( s + 1 ) > 0 formulae-sequence 𝑠 0 𝑠 1 0 {\displaystyle{\displaystyle\Re s>0,\Re(s+1)>0}} 
Zeta(s) = (1)/((1 - (2)^(1 - s))*GAMMA(s + 1))*int((exp(x)*(x)^(s))/((exp(x)+ 1)^(2)), x = 0..infinity)

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]
Failure Aborted Successful [Tested: 3] Successful [Tested: 3]
25.5.E5 ζ ⁡ ( s ) = - s ⁢ ∫ 0 ∞ x - ⌊ x ⌋ - 1 2 x s + 1 ⁢ d x Riemann-zeta 𝑠 𝑠 superscript subscript 0 𝑥 𝑥 1 2 superscript 𝑥 𝑠 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=-s\int_{0}^{\infty}\frac{x-% \left\lfloor x\right\rfloor-\frac{1}{2}}{x^{s+1}}\mathrm{d}x}}
\Riemannzeta@{s} = -s\int_{0}^{\infty}\frac{x-\floor{x}-\frac{1}{2}}{x^{s+1}}\diff{x}		 - 1 < ℜ ⁡ s , ℜ ⁡ s < 0 formulae-sequence 1 𝑠 𝑠 0 {\displaystyle{\displaystyle-1<\Re s,\Re s<0}} 
Zeta(s) = - s*int((x - floor(x)-(1)/(2))/((x)^(s + 1)), x = 0..infinity)

Zeta[s] == - s*Integrate[Divide[x - Floor[x]-Divide[1,2],(x)^(s + 1)], {x, 0, Infinity}, GenerateConditions->None]
Error Aborted -
Failed [1 / 1]
Result: Complex[-2500601.4984594644, 2.5458720000534374*^-17]
Test Values: {Rule[s, -0.5]}

25.5.E6 ζ ⁡ ( s ) = 1 2 + 1 s - 1 + 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ ( 1 e x - 1 - 1 x + 1 2 ) ⁢ x s - 1 e x ⁢ d x Riemann-zeta 𝑠 1 2 1 𝑠 1 1 Euler-Gamma 𝑠 superscript subscript 0 1 superscript 𝑒 𝑥 1 1 𝑥 1 2 superscript 𝑥 𝑠 1 superscript 𝑒 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+% \frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1% }{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{x}}\mathrm{d}x}}
\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}		 ℜ ⁡ s > - 1 , ℜ ⁡ s > 0 formulae-sequence 𝑠 1 𝑠 0 {\displaystyle{\displaystyle\Re s>-1,\Re s>0}} 
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)

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]
Failure Failure Successful [Tested: 3]
Failed [3 / 3]
Result: DirectedInfinity[]
Test Values: {Rule[s, 1.5]}


Result: Indeterminate
Test Values: {Rule[s, 0.5]}

... skip entries to safe data
25.5.E7 ζ ⁡ ( s ) = 1 2 + 1 s - 1 + ∑ m = 1 n B 2 ⁢ m ( 2 ⁢ m ) ! ⁢ ( s ) 2 ⁢ m - 1 + 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ ( 1 e x - 1 - 1 x + 1 2 - ∑ m = 1 n B 2 ⁢ m ( 2 ⁢ m ) ! ⁢ x 2 ⁢ m - 1 ) ⁢ x s - 1 e x ⁢ d x Riemann-zeta 𝑠 1 2 1 𝑠 1 superscript subscript 𝑚 1 𝑛 Bernoulli-number-B 2 𝑚 2 𝑚 Pochhammer 𝑠 2 𝑚 1 1 Euler-Gamma 𝑠 superscript subscript 0 1 superscript 𝑒 𝑥 1 1 𝑥 1 2 superscript subscript 𝑚 1 𝑛 Bernoulli-number-B 2 𝑚 2 𝑚 superscript 𝑥 2 𝑚 1 superscript 𝑥 𝑠 1 superscript 𝑒 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum% _{m=1}^{n}\frac{B_{2m}}{(2m)!}{\left(s\right)_{2m-1}}+\frac{1}{\Gamma\left(s% \right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_% {m=1}^{n}\frac{B_{2m}}{(2m)!}x^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\mathrm{d}x}}
\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}		 ℜ ⁡ s > - ( 2 ⁢ n + 1 ) , ℜ ⁡ s > 0 formulae-sequence 𝑠 2 𝑛 1 𝑠 0 {\displaystyle{\displaystyle\Re s>-(2n+1),\Re s>0}} 
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)

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]
Aborted Aborted
Failed [2 / 3]
Result: .1027534444e-1
Test Values: {s = 3/2, n = 3}


Result: .24417579e-1
Test Values: {s = 2, n = 3}

Skipped - Because timed out
25.5.E8 ζ ⁡ ( s ) = 1 2 ⁢ ( 1 - 2 - s ) ⁢ Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 sinh ⁡ x ⁢ d x Riemann-zeta 𝑠 1 2 1 superscript 2 𝑠 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2(1-2^{-s})\Gamma% \left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{\sinh x}\mathrm{d}x}}
\Riemannzeta@{s} = \frac{1}{2(1-2^{-s})\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}}{\sinh@@{x}}\diff{x}		 ℜ ⁡ s > 1 , ℜ ⁡ s > 0 formulae-sequence 𝑠 1 𝑠 0 {\displaystyle{\displaystyle\Re s>1,\Re s>0}} 
Zeta(s) = (1)/(2*(1 - (2)^(- s))*GAMMA(s))*int(((x)^(s - 1))/(sinh(x)), x = 0..infinity)

Zeta[s] == Divide[1,2*(1 - (2)^(- s))*Gamma[s]]*Integrate[Divide[(x)^(s - 1),Sinh[x]], {x, 0, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 2] Successful [Tested: 2]
25.5.E9 ζ ⁡ ( s ) = 2 s - 1 Γ ⁡ ( s + 1 ) ⁢ ∫ 0 ∞ x s ( sinh ⁡ x ) 2 ⁢ d x Riemann-zeta 𝑠 superscript 2 𝑠 1 Euler-Gamma 𝑠 1 superscript subscript 0 superscript 𝑥 𝑠 superscript 𝑥 2 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{\Gamma\left(s+1% \right)}\int_{0}^{\infty}\frac{x^{s}}{(\sinh x)^{2}}\mathrm{d}x}}
\Riemannzeta@{s} = \frac{2^{s-1}}{\EulerGamma@{s+1}}\int_{0}^{\infty}\frac{x^{s}}{(\sinh@@{x})^{2}}\diff{x}		 ℜ ⁡ s > 1 , ℜ ⁡ ( s + 1 ) > 0 formulae-sequence 𝑠 1 𝑠 1 0 {\displaystyle{\displaystyle\Re s>1,\Re(s+1)>0}} 
Zeta(s) = ((2)^(s - 1))/(GAMMA(s + 1))*int(((x)^(s))/((sinh(x))^(2)), x = 0..infinity)

Zeta[s] == Divide[(2)^(s - 1),Gamma[s + 1]]*Integrate[Divide[(x)^(s),(Sinh[x])^(2)], {x, 0, Infinity}, GenerateConditions->None]
Failure Aborted Successful [Tested: 2] Skipped - Because timed out
25.5.E10 ζ ⁡ ( s ) = 2 s - 1 1 - 2 1 - s ⁢ ∫ 0 ∞ cos ⁡ ( s ⁢ arctan ⁡ x ) ( 1 + x 2 ) s / 2 ⁢ cosh ⁡ ( 1 2 ⁢ π ⁢ x ) ⁢ d x Riemann-zeta 𝑠 superscript 2 𝑠 1 1 superscript 2 1 𝑠 superscript subscript 0 𝑠 𝑥 superscript 1 superscript 𝑥 2 𝑠 2 1 2 𝜋 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}\int_% {0}^{\infty}\frac{\cos\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}% \cosh\left(\frac{1}{2}\pi x\right)}\mathrm{d}x}}
\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}		 

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)

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]
Failure Aborted Successful [Tested: 6] Skipped - Because timed out
25.5.E11 ζ ⁡ ( s ) = 1 2 + 1 s - 1 + 2 ⁢ ∫ 0 ∞ sin ⁡ ( s ⁢ arctan ⁡ x ) ( 1 + x 2 ) s / 2 ⁢ ( e 2 ⁢ π ⁢ x - 1 ) ⁢ d x Riemann-zeta 𝑠 1 2 1 𝑠 1 2 superscript subscript 0 𝑠 𝑥 superscript 1 superscript 𝑥 2 𝑠 2 superscript 𝑒 2 𝜋 𝑥 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+2% \int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/% 2}(e^{2\pi x}-1)}\mathrm{d}x}}
\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}		 

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)

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]
Failure Successful Successful [Tested: 6] Successful [Tested: 6]
25.5.E12 ζ ⁡ ( s ) = 2 s - 1 s - 1 - 2 s ⁢ ∫ 0 ∞ sin ⁡ ( s ⁢ arctan ⁡ x ) ( 1 + x 2 ) s / 2 ⁢ ( e π ⁢ x + 1 ) ⁢ d x Riemann-zeta 𝑠 superscript 2 𝑠 1 𝑠 1 superscript 2 𝑠 superscript subscript 0 𝑠 𝑥 superscript 1 superscript 𝑥 2 𝑠 2 superscript 𝑒 𝜋 𝑥 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{s-1}-2^{s}\int_% {0}^{\infty}\frac{\sin\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^% {\pi x}+1)}\mathrm{d}x}}
\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}		 

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)

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]
Failure Successful Successful [Tested: 6] Successful [Tested: 6]
25.5.E13 ζ ⁡ ( s ) = π s / 2 s ⁢ ( s - 1 ) ⁢ Γ ⁡ ( 1 2 ⁢ s ) + π s / 2 Γ ⁡ ( 1 2 ⁢ s ) ⁢ ∫ 1 ∞ ( x s / 2 + x ( 1 - s ) / 2 ) ⁢ ω ⁢ ( x ) x ⁢ d x Riemann-zeta 𝑠 superscript 𝜋 𝑠 2 𝑠 𝑠 1 Euler-Gamma 1 2 𝑠 superscript 𝜋 𝑠 2 Euler-Gamma 1 2 𝑠 superscript subscript 1 superscript 𝑥 𝑠 2 superscript 𝑥 1 𝑠 2 𝜔 𝑥 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\pi^{s/2}}{s(s-1)\Gamma% \left(\frac{1}{2}s\right)}+\frac{\pi^{s/2}}{\Gamma\left(\frac{1}{2}s\right)}\*% \int_{1}^{\infty}\left(x^{s/2}+x^{(1-s)/2}\right)\frac{\omega(x)}{x}\mathrm{d}% x}}
\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}		 s ≠ 1 , ℜ ⁡ ( 1 2 ⁢ s ) > 0 formulae-sequence 𝑠 1 1 2 𝑠 0 {\displaystyle{\displaystyle s\neq 1,\Re(\frac{1}{2}s)>0}} 
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)

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]
Failure Aborted
Failed [30 / 30]
Result: Float(infinity)+Float(infinity)*I
Test Values: {omega = 1/2*3^(1/2)+1/2*I, s = 3/2}


Result: Float(infinity)+Float(infinity)*I
Test Values: {omega = 1/2*3^(1/2)+1/2*I, s = 1/2}

... skip entries to safe data
 Skipped - Because timed out
25.5.E14 ω ⁢ ( x ) ≡ ∑ n = 1 ∞ e - n 2 ⁢ π ⁢ x = 1 2 ⁢ ( θ 3 ⁡ ( 0 | i ⁢ x ) - 1 ) equal-by definition 𝜔 𝑥 superscript subscript 𝑛 1 superscript 𝑒 superscript 𝑛 2 𝜋 𝑥 1 2 Jacobi-theta-tau 3 0 𝑖 𝑥 1 {\displaystyle{\displaystyle\omega(x)\equiv\sum_{n=1}^{\infty}e^{-n^{2}\pi x}=% \frac{1}{2}\left(\theta_{3}\left(0\middle|ix\right)-1\right)}}
\omega(x)\defeq\sum_{n=1}^{\infty}e^{-n^{2}\pi x} = \frac{1}{2}\left(\Jacobithetatau{3}@{0}{ix}-1\right)		 

omega(x) = sum(exp(- (n)^(2)* Pi*x), n = 1..infinity) = (1)/(2)*(JacobiTheta3(0,exp(I*Pi*I*x))- 1)

\[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)
Failure Failure Error
Failed [30 / 30]
Result: Plus[-0.008983297533541545, False]
Test Values: {Rule[x, 1.5], Rule[ω, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}


Result: Plus[-0.008983297533541545, False]
Test Values: {Rule[x, 1.5], Rule[ω, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
25.5.E15 ζ ⁡ ( s ) = 1 s - 1 + sin ⁡ ( π ⁢ s ) π ⁢ ∫ 0 ∞ ( ln ⁡ ( 1 + x ) - ψ ⁡ ( 1 + x ) ) ⁢ x - s ⁢ d x Riemann-zeta 𝑠 1 𝑠 1 𝜋 𝑠 𝜋 superscript subscript 0 1 𝑥 digamma 1 𝑥 superscript 𝑥 𝑠 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\left(% \pi s\right)}{\pi}\*\int_{0}^{\infty}(\ln\left(1+x\right)-\psi\left(1+x\right)% )x^{-s}\mathrm{d}x}}
\Riemannzeta@{s} = \frac{1}{s-1}+\frac{\sin@{\pi s}}{\pi}\*\int_{0}^{\infty}(\ln@{1+x}-\digamma@{1+x})x^{-s}\diff{x}		 

Zeta(s) = (1)/(s - 1)+(sin(Pi*s))/(Pi)* int((ln(1 + x)- Psi(1 + x))*(x)^(- s), x = 0..infinity)

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]
Failure Aborted
Failed [5 / 6]
Result: Float(-infinity)
Test Values: {s = -3/2}


Result: Float(infinity)
Test Values: {s = 3/2}

... skip entries to safe data
 Skipped - Because timed out
25.5.E16 ζ ⁡ ( s ) = 1 s - 1 + sin ⁡ ( π ⁢ s ) π ⁢ ( s - 1 ) ⁢ ∫ 0 ∞ ( 1 1 + x - ψ ′ ⁡ ( 1 + x ) ) ⁢ x 1 - s ⁢ d x Riemann-zeta 𝑠 1 𝑠 1 𝜋 𝑠 𝜋 𝑠 1 superscript subscript 0 1 1 𝑥 diffop digamma 1 1 𝑥 superscript 𝑥 1 𝑠 𝑥 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\left(% \pi s\right)}{\pi(s-1)}\*\int_{0}^{\infty}\left(\frac{1}{1+x}-\psi'\left(1+x% \right)\right)x^{1-s}\mathrm{d}x}}
\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}		 

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)

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]
Failure Aborted
Failed [4 / 6]
Result: Float(-infinity)
Test Values: {s = -3/2}


Result: Float(infinity)
Test Values: {s = -1/2}

... skip entries to safe data
 Skipped - Because timed out
25.5.E17 ζ ⁡ ( 1 + s ) = sin ⁡ ( π ⁢ s ) π ⁢ ∫ 0 ∞ ( γ + ψ ⁡ ( 1 + x ) ) ⁢ x - s - 1 ⁢ d x Riemann-zeta 1 𝑠 𝜋 𝑠 𝜋 superscript subscript 0 digamma 1 𝑥 superscript 𝑥 𝑠 1 𝑥 {\displaystyle{\displaystyle\zeta\left(1+s\right)=\frac{\sin\left(\pi s\right)% }{\pi}\int_{0}^{\infty}\left(\gamma+\psi\left(1+x\right)\right)x^{-s-1}\mathrm% {d}x}}
\Riemannzeta@{1+s} = \frac{\sin@{\pi s}}{\pi}\int_{0}^{\infty}\left(\EulerConstant+\digamma@{1+x}\right)x^{-s-1}\diff{x}		 

Zeta(1 + s) = (sin(Pi*s))/(Pi)*int((gamma + Psi(1 + x))*(x)^(- s - 1), x = 0..infinity)

Zeta[1 + s] == Divide[Sin[Pi*s],Pi]*Integrate[(EulerGamma + PolyGamma[1 + x])*(x)^(- s - 1), {x, 0, Infinity}, GenerateConditions->None]
Failure Failure
Failed [6 / 6]
Result: Float(-infinity)
Test Values: {s = -3/2}


Result: Float(-infinity)
Test Values: {s = 3/2}

... skip entries to safe data
 Skipped - Because timed out
25.5.E18 ζ ⁡ ( 1 + s ) = sin ⁡ ( π ⁢ s ) π ⁢ s ⁢ ∫ 0 ∞ ψ ′ ⁡ ( 1 + x ) ⁢ x - s ⁢ d x Riemann-zeta 1 𝑠 𝜋 𝑠 𝜋 𝑠 superscript subscript 0 diffop digamma 1 1 𝑥 superscript 𝑥 𝑠 𝑥 {\displaystyle{\displaystyle\zeta\left(1+s\right)=\frac{\sin\left(\pi s\right)% }{\pi s}\int_{0}^{\infty}\psi'\left(1+x\right)x^{-s}\mathrm{d}x}}
\Riemannzeta@{1+s} = \frac{\sin@{\pi s}}{\pi s}\int_{0}^{\infty}\digamma'@{1+x}x^{-s}\diff{x}		 

Zeta(1 + s) = (sin(Pi*s))/(Pi*s)*int(subs( temp=1 + x, diff( Psi(temp), temp$(1) ) )*(x)^(- s), x = 0..infinity)

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]
Failure Failure
Failed [5 / 6]
Result: Float(infinity)
Test Values: {s = -3/2}


Result: Float(infinity)
Test Values: {s = 3/2}

... skip entries to safe data
Failed [5 / 6]
Result: Complex[1.225514223589735984806690657246494542888138*^+10484, 2.5458720000534374*^-17]
Test Values: {Rule[s, -1.5]}


Result: 2.9996624622276097*^37
Test Values: {Rule[s, 1.5]}

... skip entries to safe data
25.5.E19 ζ ⁡ ( m + s ) = ( - 1 ) m - 1 ⁢ Γ ⁡ ( s ) ⁢ sin ⁡ ( π ⁢ s ) π ⁢ Γ ⁡ ( m + s ) ⁢ ∫ 0 ∞ ψ ( m ) ⁡ ( 1 + x ) ⁢ x - s ⁢ d x Riemann-zeta 𝑚 𝑠 superscript 1 𝑚 1 Euler-Gamma 𝑠 𝜋 𝑠 𝜋 Euler-Gamma 𝑚 𝑠 superscript subscript 0 digamma 𝑚 1 𝑥 superscript 𝑥 𝑠 𝑥 {\displaystyle{\displaystyle\zeta\left(m+s\right)=(-1)^{m-1}\frac{\Gamma\left(% s\right)\sin\left(\pi s\right)}{\pi\Gamma\left(m+s\right)}\*\int_{0}^{\infty}{% \psi^{(m)}}\left(1+x\right)x^{-s}\mathrm{d}x}}
\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}		 ℜ ⁡ s > 0 , ℜ ⁡ ( m + s ) > 0 formulae-sequence 𝑠 0 𝑚 𝑠 0 {\displaystyle{\displaystyle\Re s>0,\Re(m+s)>0}} 
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)

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]
Failure Failure
Failed [2 / 3]
Result: Float(infinity)
Test Values: {s = 3/2, m = 3}


Result: Float(-infinity)
Test Values: {s = 2, m = 3}

Skipped - Because timed out
25.5.E20 ζ ⁡ ( s ) = Γ ⁡ ( 1 - s ) 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) z s - 1 e - z - 1 ⁢ d z Riemann-zeta 𝑠 Euler-Gamma 1 𝑠 2 𝜋 𝑖 superscript subscript limit-from 0 superscript 𝑧 𝑠 1 superscript 𝑒 𝑧 1 𝑧 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{% 2\pi i}\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}-1}\mathrm{d}z}}
\Riemannzeta@{s} = \frac{\EulerGamma@{1-s}}{2\pi i}\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}-1}\diff{z}		 ℜ ⁡ ( 1 - s ) > 0 1 𝑠 0 {\displaystyle{\displaystyle\Re(1-s)>0}} 
Zeta(s) = (GAMMA(1 - s))/(2*Pi*I)*int(((z)^(s - 1))/(exp(- z)- 1), z = - infinity..(0 +))

Zeta[s] == Divide[Gamma[1 - s],2*Pi*I]*Integrate[Divide[(z)^(s - 1),Exp[- z]- 1], {z, - Infinity, (0 +)}, GenerateConditions->None]
Error Failure - Error
25.5.E21 ζ ⁡ ( s ) = Γ ⁡ ( 1 - s ) 2 ⁢ π ⁢ i ⁢ ( 1 - 2 1 - s ) ⁢ ∫ - ∞ ( 0 + ) z s - 1 e - z + 1 ⁢ d z Riemann-zeta 𝑠 Euler-Gamma 1 𝑠 2 𝜋 𝑖 1 superscript 2 1 𝑠 superscript subscript limit-from 0 superscript 𝑧 𝑠 1 superscript 𝑒 𝑧 1 𝑧 {\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{% 2\pi i(1-2^{1-s})}\*\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}+1}\mathrm{d}z}}
\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}		 ℜ ⁡ ( 1 - s ) > 0 1 𝑠 0 {\displaystyle{\displaystyle\Re(1-s)>0}} 
Zeta(s) = (GAMMA(1 - s))/(2*Pi*I*(1 - (2)^(1 - s)))* int(((z)^(s - 1))/(exp(- z)+ 1), z = - infinity..(0 +))

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]
Error Failure - Error
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