Zeta and Related Functions - 25.11 Hurwitz Zeta Function

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DLMF Formula Constraints Maple Mathematica Symbolic
Maple 		Symbolic
Mathematica 		Numeric
Maple 		Numeric
Mathematica
25.11.E1 ζ ⁡ ( s , a ) = ∑ n = 0 ∞ 1 ( n + a ) s Hurwitz-zeta 𝑠 𝑎 superscript subscript 𝑛 0 1 superscript 𝑛 𝑎 𝑠 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{1}{% (n+a)^{s}}}}
\Hurwitzzeta@{s}{a} = \sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}		 ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}} 
Zeta(0, s, a) = sum((1)/((n + a)^(s)), n = 0..infinity)

HurwitzZeta[s, a] == Sum[Divide[1,(n + a)^(s)], {n, 0, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 2] Successful [Tested: 2]
25.11.E2 ζ ⁡ ( s , 1 ) = ζ ⁡ ( s ) Hurwitz-zeta 𝑠 1 Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(s,1\right)=\zeta\left(s\right)}}
\Hurwitzzeta@{s}{1} = \Riemannzeta@{s}		 

Zeta(0, s, 1) = Zeta(s)

HurwitzZeta[s, 1] == Zeta[s]
Successful Successful - Successful [Tested: 6]
25.11.E3 ζ ⁡ ( s , a ) = ζ ⁡ ( s , a + 1 ) + a - s Hurwitz-zeta 𝑠 𝑎 Hurwitz-zeta 𝑠 𝑎 1 superscript 𝑎 𝑠 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\zeta\left(s,a+1\right)+a^{-% s}}}
\Hurwitzzeta@{s}{a} = \Hurwitzzeta@{s}{a+1}+a^{-s}		 

Zeta(0, s, a) = Zeta(0, s, a + 1)+ (a)^(- s)

HurwitzZeta[s, a] == HurwitzZeta[s, a + 1]+ (a)^(- s)
Failure Successful Error
Failed [3 / 36]
Result: Indeterminate
Test Values: {Rule[a, -2], Rule[s, 1.5]}


Result: Indeterminate
Test Values: {Rule[a, -2], Rule[s, 0.5]}

... skip entries to safe data
25.11.E4 ζ ⁡ ( s , a ) = ζ ⁡ ( s , a + m ) + ∑ n = 0 m - 1 1 ( n + a ) s Hurwitz-zeta 𝑠 𝑎 Hurwitz-zeta 𝑠 𝑎 𝑚 superscript subscript 𝑛 0 𝑚 1 1 superscript 𝑛 𝑎 𝑠 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\zeta\left(s,a+m\right)+\sum% _{n=0}^{m-1}\frac{1}{(n+a)^{s}}}}
\Hurwitzzeta@{s}{a} = \Hurwitzzeta@{s}{a+m}+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{s}}		 

Zeta(0, s, a) = Zeta(0, s, a + m)+ sum((1)/((n + a)^(s)), n = 0..m - 1)

HurwitzZeta[s, a] == HurwitzZeta[s, a + m]+ Sum[Divide[1,(n + a)^(s)], {n, 0, m - 1}, GenerateConditions->None]
Failure Successful Error Successful [Tested: 36]
25.11.E5 ζ ⁡ ( s , a ) = ∑ n = 0 N 1 ( n + a ) s + ( N + a ) 1 - s s - 1 - s ⁢ ∫ N ∞ x - ⌊ x ⌋ ( x + a ) s + 1 ⁢ d x Hurwitz-zeta 𝑠 𝑎 superscript subscript 𝑛 0 𝑁 1 superscript 𝑛 𝑎 𝑠 superscript 𝑁 𝑎 1 𝑠 𝑠 1 𝑠 superscript subscript 𝑁 𝑥 𝑥 superscript 𝑥 𝑎 𝑠 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{N}\frac{1}{(n+a)% ^{s}}+\frac{(N+a)^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right% \rfloor}{(x+a)^{s+1}}\mathrm{d}x}}
\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}		 s ≠ 1 , ℜ ⁡ s > 0 , a > 0 formulae-sequence 𝑠 1 formulae-sequence 𝑠 0 𝑎 0 {\displaystyle{\displaystyle s\neq 1,\Re s>0,a>0}} 
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)

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]
Failure Aborted
Failed [8 / 9]
Result: .2257548023
Test Values: {a = 3/2, s = 3/2, N = 3}


Result: Float(infinity)
Test Values: {a = 3/2, s = 1/2, N = 3}

... skip entries to safe data
 Skipped - Because timed out
25.11.E8 ζ ⁡ ( s , 1 2 ⁢ a ) = ζ ⁡ ( s , 1 2 ⁢ a + 1 2 ) + 2 s ⁢ ∑ n = 0 ∞ ( - 1 ) n ( n + a ) s Hurwitz-zeta 𝑠 1 2 𝑎 Hurwitz-zeta 𝑠 1 2 𝑎 1 2 superscript 2 𝑠 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑛 𝑎 𝑠 {\displaystyle{\displaystyle\zeta\left(s,\tfrac{1}{2}a\right)=\zeta\left(s,% \tfrac{1}{2}a+\tfrac{1}{2}\right)+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a% )^{s}}}}
\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}}		 ℜ ⁡ s > 0 , s ≠ 1 , 0 < a , a ≤ 1 formulae-sequence 𝑠 0 formulae-sequence 𝑠 1 formulae-sequence 0 𝑎 𝑎 1 {\displaystyle{\displaystyle\Re s>0,s\neq 1,0<a,a\leq 1}} 
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)

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]
Failure Successful Successful [Tested: 3] Successful [Tested: 3]
25.11.E9 ζ ⁡ ( 1 - s , a ) = 2 ⁢ Γ ⁡ ( s ) ( 2 ⁢ π ) s ⁢ ∑ n = 1 ∞ 1 n s ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ s - 2 ⁢ n ⁢ π ⁢ a ) Hurwitz-zeta 1 𝑠 𝑎 2 Euler-Gamma 𝑠 superscript 2 𝜋 𝑠 superscript subscript 𝑛 1 1 superscript 𝑛 𝑠 1 2 𝜋 𝑠 2 𝑛 𝜋 𝑎 {\displaystyle{\displaystyle\zeta\left(1-s,a\right)=\frac{2\Gamma\left(s\right% )}{(2\pi)^{s}}\*\sum_{n=1}^{\infty}\frac{1}{n^{s}}\cos\left(\tfrac{1}{2}\pi s-% 2n\pi a\right)}}
\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}		 ℜ ⁡ s > 1 , 0 < a , a ≤ 1 , ℜ ⁡ s > 0 formulae-sequence 𝑠 1 formulae-sequence 0 𝑎 formulae-sequence 𝑎 1 𝑠 0 {\displaystyle{\displaystyle\Re s>1,0<a,a\leq 1,\Re s>0}} 
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)

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]
Error Failure - Skip - No test values generated
25.11.E10 ζ ⁡ ( s , a ) = ∑ n = 0 ∞ ( s ) n n ! ⁢ ζ ⁡ ( n + s ) ⁢ ( 1 - a ) n Hurwitz-zeta 𝑠 𝑎 superscript subscript 𝑛 0 Pochhammer 𝑠 𝑛 𝑛 Riemann-zeta 𝑛 𝑠 superscript 1 𝑎 𝑛 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{{% \left(s\right)_{n}}}{n!}\zeta\left(n+s\right)(1-a)^{n}}}
\Hurwitzzeta@{s}{a} = \sum_{n=0}^{\infty}\frac{\Pochhammersym{s}{n}}{n!}\Riemannzeta@{n+s}(1-a)^{n}		 s ≠ 1 , | a - 1 | < 1 formulae-sequence 𝑠 1 𝑎 1 1 {\displaystyle{\displaystyle s\neq 1,|a-1|<1}} 
Zeta(0, s, a) = sum((pochhammer(s, n))/(factorial(n))*Zeta(n + s)*(1 - a)^(n), n = 0..infinity)

HurwitzZeta[s, a] == Sum[Divide[Pochhammer[s, n],(n)!]*Zeta[n + s]*(1 - a)^(n), {n, 0, Infinity}, GenerateConditions->None]
Error Aborted -
Failed [2 / 2]
Result: Plus[2.612375348685488, Times[-1.0, NSum[Times[Power[0, n], Power[Factorial[n], -1], Pochhammer[1.5, n], Zeta[Plus[1.5, n]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, 1], Rule[s, 1.5]}


Result: Plus[1.6449340668482262, Times[-1.0, NSum[Times[Power[0, n], Power[Factorial[n], -1], Pochhammer[2, n], Zeta[Plus[2, n]]]
Test Values: {n, 0, DirectedInfinity[1]}, Rule[GenerateConditions, None]]]], {Rule[a, 1], Rule[s, 2]}

25.11.E11 ζ ⁡ ( s , 1 2 ) = ( 2 s - 1 ) ⁢ ζ ⁡ ( s ) Hurwitz-zeta 𝑠 1 2 superscript 2 𝑠 1 Riemann-zeta 𝑠 {\displaystyle{\displaystyle\zeta\left(s,\tfrac{1}{2}\right)=(2^{s}-1)\zeta% \left(s\right)}}
\Hurwitzzeta@{s}{\tfrac{1}{2}} = (2^{s}-1)\Riemannzeta@{s}		 s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}} 
Zeta(0, s, (1)/(2)) = ((2)^(s)- 1)*Zeta(s)

HurwitzZeta[s, Divide[1,2]] == ((2)^(s)- 1)*Zeta[s]
Successful Successful - Successful [Tested: 6]
25.11.E12 ζ ⁡ ( n + 1 , a ) = ( - 1 ) n + 1 ⁢ ψ ( n ) ⁡ ( a ) n ! Hurwitz-zeta 𝑛 1 𝑎 superscript 1 𝑛 1 digamma 𝑛 𝑎 𝑛 {\displaystyle{\displaystyle\zeta\left(n+1,a\right)=\frac{(-1)^{n+1}{\psi^{(n)% }}\left(a\right)}{n!}}}
\Hurwitzzeta@{n+1}{a} = \frac{(-1)^{n+1}\digamma^{(n)}@{a}}{n!}		 

Zeta(0, n + 1, a) = ((- 1)^(n + 1)* diff( Psi(a), a$(n) ))/(factorial(n))

HurwitzZeta[n + 1, a] == Divide[(- 1)^(n + 1)* D[PolyGamma[a], {a, n}],(n)!]
Failure Failure Error Successful [Tested: 1]
25.11.E13 ζ ⁡ ( 0 , a ) = 1 2 - a Hurwitz-zeta 0 𝑎 1 2 𝑎 {\displaystyle{\displaystyle\zeta\left(0,a\right)=\tfrac{1}{2}-a}}
\Hurwitzzeta@{0}{a} = \tfrac{1}{2}-a		 

Zeta(0, 0, a) = (1)/(2)- a

HurwitzZeta[0, a] == Divide[1,2]- a
Successful Successful - Successful [Tested: 6]
25.11.E14 ζ ⁡ ( - n , a ) = - B n + 1 ⁡ ( a ) n + 1 Hurwitz-zeta 𝑛 𝑎 Bernoulli-polynomial-B 𝑛 1 𝑎 𝑛 1 {\displaystyle{\displaystyle\zeta\left(-n,a\right)=-\frac{B_{n+1}\left(a\right% )}{n+1}}}
\Hurwitzzeta@{-n}{a} = -\frac{\BernoullipolyB{n+1}@{a}}{n+1}		 

Zeta(0, - n, a) = -(bernoulli(n + 1, a))/(n + 1)

HurwitzZeta[- n, a] == -Divide[BernoulliB[n + 1, a],n + 1]
Successful Failure - Successful [Tested: 1]
25.11.E15 ζ ⁡ ( s , k ⁢ a ) = k - s ⁢ ∑ n = 0 k - 1 ζ ⁡ ( s , a + n k ) Hurwitz-zeta 𝑠 𝑘 𝑎 superscript 𝑘 𝑠 superscript subscript 𝑛 0 𝑘 1 Hurwitz-zeta 𝑠 𝑎 𝑛 𝑘 {\displaystyle{\displaystyle\zeta\left(s,ka\right)=k^{-s}\*\sum_{n=0}^{k-1}% \zeta\left(s,a+\frac{n}{k}\right)}}
\Hurwitzzeta@{s}{ka} = k^{-s}\*\sum_{n=0}^{k-1}\Hurwitzzeta@{s}{a+\frac{n}{k}}		 s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}} 
Zeta(0, s, k*a) = (k)^(- s)* sum(Zeta(0, s, a +(n)/(k)), n = 0..k - 1)

HurwitzZeta[s, k*a] == (k)^(- s)* Sum[HurwitzZeta[s, a +Divide[n,k]], {n, 0, k - 1}, GenerateConditions->None]
Failure Failure Error
Failed [2 / 2]
Result: 1.3535533905932735
Test Values: {Rule[a, 1], Rule[k, 3], Rule[Times[a, k], 1], Rule[s, 1.5]}


Result: 1.2499999999999998
Test Values: {Rule[a, 1], Rule[k, 3], Rule[Times[a, k], 1], Rule[s, 2]}

25.11.E16 ζ ⁡ ( 1 - s , h k ) = 2 ⁢ Γ ⁡ ( s ) ( 2 ⁢ π ⁢ k ) s ⁢ ∑ r = 1 k cos ⁡ ( π ⁢ s 2 - 2 ⁢ π ⁢ r ⁢ h k ) ⁢ ζ ⁡ ( s , r k ) Hurwitz-zeta 1 𝑠 ℎ 𝑘 2 Euler-Gamma 𝑠 superscript 2 𝜋 𝑘 𝑠 superscript subscript 𝑟 1 𝑘 𝜋 𝑠 2 2 𝜋 𝑟 ℎ 𝑘 Hurwitz-zeta 𝑠 𝑟 𝑘 {\displaystyle{\displaystyle\zeta\left(1-s,\frac{h}{k}\right)=\frac{2\Gamma% \left(s\right)}{(2\pi k)^{s}}\*\sum_{r=1}^{k}\cos\left(\frac{\pi s}{2}-\frac{2% \pi rh}{k}\right)\zeta\left(s,\frac{r}{k}\right)}}
\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}}		 1 ≤ h , h ≤ k , ℜ ⁡ s > 0 formulae-sequence 1 ℎ formulae-sequence ℎ 𝑘 𝑠 0 {\displaystyle{\displaystyle 1\leq h,h\leq k,\Re s>0}} 
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)

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]
Error Aborted - Skipped - Because timed out
25.11.E17 ∂ ∂ ⁡ a ⁡ ζ ⁡ ( s , a ) = - s ⁢ ζ ⁡ ( s + 1 , a ) partial-derivative 𝑎 Hurwitz-zeta 𝑠 𝑎 𝑠 Hurwitz-zeta 𝑠 1 𝑎 {\displaystyle{\displaystyle\frac{\partial}{\partial a}\zeta\left(s,a\right)=-% s\zeta\left(s+1,a\right)}}
\pderiv{}{a}\Hurwitzzeta@{s}{a} = -s\Hurwitzzeta@{s+1}{a}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
diff(Zeta(0, s, a), a) = - s*Zeta(0, s + 1, a)

D[HurwitzZeta[s, a], a] == - s*HurwitzZeta[s + 1, a]
Error Successful - Successful [Tested: 3]
25.11.E18 ζ ′ ⁡ ( 0 , a ) = ln ⁡ Γ ⁡ ( a ) - 1 2 ⁢ ln ⁡ ( 2 ⁢ π ) diffop Hurwitz-zeta 1 0 𝑎 Euler-Gamma 𝑎 1 2 2 𝜋 {\displaystyle{\displaystyle\zeta'\left(0,a\right)=\ln\Gamma\left(a\right)-% \tfrac{1}{2}\ln\left(2\pi\right)}}
\Hurwitzzeta'@{0}{a} = \ln@@{\EulerGamma@{a}}-\tfrac{1}{2}\ln@{2\pi}		 a > 0 , ℜ ⁡ a > 0 formulae-sequence 𝑎 0 𝑎 0 {\displaystyle{\displaystyle a>0,\Re a>0}} 
subs( temp=0, diff( Zeta(0, temp, a), temp$(1) ) ) = ln(GAMMA(a))-(1)/(2)*ln(2*Pi)

(D[HurwitzZeta[temp, a], {temp, 1}]/.temp-> 0) == Log[Gamma[a]]-Divide[1,2]*Log[2*Pi]
Failure Failure Successful [Tested: 3] Successful [Tested: 1]
25.11.E21 ζ ′ ⁡ ( 1 - 2 ⁢ n , h k ) = ( ψ ⁡ ( 2 ⁢ n ) - ln ⁡ ( 2 ⁢ π ⁢ k ) ) ⁢ B 2 ⁢ n ⁡ ( h / k ) 2 ⁢ n - ( ψ ⁡ ( 2 ⁢ n ) - ln ⁡ ( 2 ⁢ π ) ) ⁢ B 2 ⁢ n 2 ⁢ n ⁢ k 2 ⁢ n + ( - 1 ) n + 1 ⁢ π ( 2 ⁢ π ⁢ k ) 2 ⁢ n ⁢ ∑ r = 1 k - 1 sin ⁡ ( 2 ⁢ π ⁢ r ⁢ h k ) ⁢ ψ ( 2 ⁢ n - 1 ) ⁡ ( r k ) + ( - 1 ) n + 1 ⁢ 2 ⋅ ( 2 ⁢ n - 1 ) ! ( 2 ⁢ π ⁢ k ) 2 ⁢ n ⁢ ∑ r = 1 k - 1 cos ⁡ ( 2 ⁢ π ⁢ r ⁢ h k ) ⁢ ζ ′ ⁡ ( 2 ⁢ n , r k ) + ζ ′ ⁡ ( 1 - 2 ⁢ n ) k 2 ⁢ n diffop Hurwitz-zeta 1 1 2 𝑛 ℎ 𝑘 digamma 2 𝑛 2 𝜋 𝑘 Bernoulli-polynomial-B 2 𝑛 ℎ 𝑘 2 𝑛 digamma 2 𝑛 2 𝜋 Bernoulli-number-B 2 𝑛 2 𝑛 superscript 𝑘 2 𝑛 superscript 1 𝑛 1 𝜋 superscript 2 𝜋 𝑘 2 𝑛 superscript subscript 𝑟 1 𝑘 1 2 𝜋 𝑟 ℎ 𝑘 digamma 2 𝑛 1 𝑟 𝑘 ⋅ superscript 1 𝑛 1 2 2 𝑛 1 superscript 2 𝜋 𝑘 2 𝑛 superscript subscript 𝑟 1 𝑘 1 2 𝜋 𝑟 ℎ 𝑘 diffop Hurwitz-zeta 1 2 𝑛 𝑟 𝑘 diffop Riemann-zeta 1 1 2 𝑛 superscript 𝑘 2 𝑛 {\displaystyle{\displaystyle\zeta'\left(1-2n,\frac{h}{k}\right)=\frac{(\psi% \left(2n\right)-\ln\left(2\pi k\right))B_{2n}\left(h/k\right)}{2n}-\frac{(\psi% \left(2n\right)-\ln\left(2\pi\right))B_{2n}}{2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2% \pi k)^{2n}}\sum_{r=1}^{k-1}\sin\left(\frac{2\pi rh}{k}\right){\psi^{(2n-1)}}% \left(\frac{r}{k}\right)+\frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=% 1}^{k-1}\cos\left(\frac{2\pi rh}{k}\right)\zeta'\left(2n,\frac{r}{k}\right)+% \frac{\zeta'\left(1-2n\right)}{k^{2n}}}}
\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}}		 

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))

(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)]
Failure Aborted
Failed [70 / 90]
Result: -.2303130415-.107731247e-1*I
Test Values: {h = 1/2*3^(1/2)+1/2*I, k = 1, n = 1}


Result: .8722916351e-1-.251419603e-1*I
Test Values: {h = 1/2*3^(1/2)+1/2*I, k = 1, n = 2}

... skip entries to safe data
 Skipped - Because timed out
25.11.E22 ζ ′ ⁡ ( 1 - 2 ⁢ n , 1 2 ) = - B 2 ⁢ n ⁢ ln ⁡ 2 n ⋅ 4 n - ( 2 2 ⁢ n - 1 - 1 ) ⁢ ζ ′ ⁡ ( 1 - 2 ⁢ n ) 2 2 ⁢ n - 1 diffop Hurwitz-zeta 1 1 2 𝑛 1 2 Bernoulli-number-B 2 𝑛 2 ⋅ 𝑛 superscript 4 𝑛 superscript 2 2 𝑛 1 1 diffop Riemann-zeta 1 1 2 𝑛 superscript 2 2 𝑛 1 {\displaystyle{\displaystyle\zeta'\left(1-2n,\tfrac{1}{2}\right)=-\frac{B_{2n}% \ln 2}{n\cdot 4^{n}}-\frac{(2^{2n-1}-1)\zeta'\left(1-2n\right)}{2^{2n-1}}}}
\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}}		 

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))

(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)]
Failure Failure Successful [Tested: 1] Successful [Tested: 1]
25.11.E23 ζ ′ ⁡ ( 1 - 2 ⁢ n , 1 3 ) = - π ⁢ ( 9 n - 1 ) ⁢ B 2 ⁢ n 8 ⁢ n ⁢ 3 ⁢ ( 3 2 ⁢ n - 1 - 1 ) - B 2 ⁢ n ⁢ ln ⁡ 3 4 ⁢ n ⋅ 3 2 ⁢ n - 1 - ( - 1 ) n ⁢ ψ ( 2 ⁢ n - 1 ) ⁡ ( 1 3 ) 2 ⁢ 3 ⁢ ( 6 ⁢ π ) 2 ⁢ n - 1 - ( 3 2 ⁢ n - 1 - 1 ) ⁢ ζ ′ ⁡ ( 1 - 2 ⁢ n ) 2 ⋅ 3 2 ⁢ n - 1 diffop Hurwitz-zeta 1 1 2 𝑛 1 3 𝜋 superscript 9 𝑛 1 Bernoulli-number-B 2 𝑛 8 𝑛 3 superscript 3 2 𝑛 1 1 Bernoulli-number-B 2 𝑛 3 ⋅ 4 𝑛 superscript 3 2 𝑛 1 superscript 1 𝑛 digamma 2 𝑛 1 1 3 2 3 superscript 6 𝜋 2 𝑛 1 superscript 3 2 𝑛 1 1 diffop Riemann-zeta 1 1 2 𝑛 ⋅ 2 superscript 3 2 𝑛 1 {\displaystyle{\displaystyle\zeta'\left(1-2n,\tfrac{1}{3}\right)=-\frac{\pi(9^% {n}-1)B_{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{B_{2n}\ln 3}{4n\cdot 3^{2n-1}}-% \frac{(-1)^{n}{\psi^{(2n-1)}}\left(\frac{1}{3}\right)}{2\sqrt{3}(6\pi)^{2n-1}}% -\frac{\left(3^{2n-1}-1\right)\zeta'\left(1-2n\right)}{2\cdot 3^{2n-1}}}}
\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}}		 

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))

(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)]
Failure Failure Successful [Tested: 1]
Failed [1 / 1]
Result: Plus[0.010637344739107386, Times[-1.2131199967624389*^-7, D[-3.1320337800208065
Test Values: {0.3333333333333333, 5.0}]]], {Rule[a, 1], Rule[n, 3]}

25.11.E24 ∑ r = 1 k - 1 ζ ′ ⁡ ( s , r k ) = ( k s - 1 ) ⁢ ζ ′ ⁡ ( s ) + k s ⁢ ζ ⁡ ( s ) ⁢ ln ⁡ k superscript subscript 𝑟 1 𝑘 1 diffop Hurwitz-zeta 1 𝑠 𝑟 𝑘 superscript 𝑘 𝑠 1 diffop Riemann-zeta 1 𝑠 superscript 𝑘 𝑠 Riemann-zeta 𝑠 𝑘 {\displaystyle{\displaystyle\sum_{r=1}^{k-1}\zeta'\left(s,\frac{r}{k}\right)=(% k^{s}-1)\zeta'\left(s\right)+k^{s}\zeta\left(s\right)\ln k}}
\sum_{r=1}^{k-1}\Hurwitzzeta'@{s}{\frac{r}{k}} = (k^{s}-1)\Riemannzeta'@{s}+k^{s}\Riemannzeta@{s}\ln@@{k}		 s ≠ 1 𝑠 1 {\displaystyle{\displaystyle s\neq 1}} 
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)

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]
Failure Failure Successful [Tested: 6]
Failed [1 / 1]
Result: Indeterminate
Test Values: {Rule[a, 1], Rule[k, 3], Rule[s, 1]}

25.11.E25 ζ ⁡ ( s , a ) = 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 ⁢ e - a ⁢ x 1 - e - x ⁢ d x Hurwitz-zeta 𝑠 𝑎 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 superscript 𝑒 𝑎 𝑥 1 superscript 𝑒 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{\Gamma\left(s\right% )}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-e^{-x}}\mathrm{d}x}}
\Hurwitzzeta@{s}{a} = \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1-e^{-x}}\diff{x}		 ℜ ⁡ s > 1 , ℜ ⁡ a > 0 , ℜ ⁡ s > 0 formulae-sequence 𝑠 1 formulae-sequence 𝑎 0 𝑠 0 {\displaystyle{\displaystyle\Re s>1,\Re a>0,\Re s>0}} 
Zeta(0, s, a) = (1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 - exp(- x)), x = 0..infinity)

HurwitzZeta[s, a] == Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 - Exp[- x]], {x, 0, Infinity}, GenerateConditions->None]
Failure Successful Successful [Tested: 6] Successful [Tested: 6]
25.11.E26 ζ ⁡ ( s , a ) = - s ⁢ ∫ - a ∞ x - ⌊ x ⌋ - 1 2 ( x + a ) s + 1 ⁢ d x Hurwitz-zeta 𝑠 𝑎 𝑠 superscript subscript 𝑎 𝑥 𝑥 1 2 superscript 𝑥 𝑎 𝑠 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=-s\int_{-a}^{\infty}\frac{x-% \left\lfloor x\right\rfloor-\frac{1}{2}}{(x+a)^{s+1}}\mathrm{d}x}}
\Hurwitzzeta@{s}{a} = -s\int_{-a}^{\infty}\frac{x-\floor{x}-\frac{1}{2}}{(x+a)^{s+1}}\diff{x}		 - 1 < ℜ ⁡ s , ℜ ⁡ s < 0 , 0 < a , a ≤ 1 formulae-sequence 1 𝑠 formulae-sequence 𝑠 0 formulae-sequence 0 𝑎 𝑎 1 {\displaystyle{\displaystyle-1<\Re s,\Re s<0,0<a,a\leq 1}} 
Zeta(0, s, a) = - s*int((x - floor(x)-(1)/(2))/((x + a)^(s + 1)), x = - a..infinity)

HurwitzZeta[s, a] == - s*Integrate[Divide[x - Floor[x]-Divide[1,2],(x + a)^(s + 1)], {x, - a, Infinity}, GenerateConditions->None]
Error Aborted - Skip - No test values generated
25.11.E27 ζ ⁡ ( s , a ) = 1 2 ⁢ a - s + a 1 - s s - 1 + 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ ( 1 e x - 1 - 1 x + 1 2 ) ⁢ x s - 1 e a ⁢ x ⁢ d x Hurwitz-zeta 𝑠 𝑎 1 2 superscript 𝑎 𝑠 superscript 𝑎 1 𝑠 𝑠 1 1 Euler-Gamma 𝑠 superscript subscript 0 1 superscript 𝑒 𝑥 1 1 𝑥 1 2 superscript 𝑥 𝑠 1 superscript 𝑒 𝑎 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1% -s}}{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^{ax}}\mathrm{d}x}}
\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}		 ℜ ⁡ s > - 1 , s ≠ 1 , ℜ ⁡ a > 0 , ℜ ⁡ s > 0 formulae-sequence 𝑠 1 formulae-sequence 𝑠 1 formulae-sequence 𝑎 0 𝑠 0 {\displaystyle{\displaystyle\Re s>-1,s\neq 1,\Re a>0,\Re s>0}} 
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)

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]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
25.11.E28 ζ ⁡ ( s , a ) = 1 2 ⁢ a - s + a 1 - s s - 1 + ∑ k = 1 n B 2 ⁢ k ( 2 ⁢ k ) ! ⁢ ( s ) 2 ⁢ k - 1 ⁢ a 1 - s - 2 ⁢ k + 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ ( 1 e x - 1 - 1 x + 1 2 - ∑ k = 1 n B 2 ⁢ k ( 2 ⁢ k ) ! ⁢ x 2 ⁢ k - 1 ) ⁢ x s - 1 ⁢ e - a ⁢ x ⁢ d x Hurwitz-zeta 𝑠 𝑎 1 2 superscript 𝑎 𝑠 superscript 𝑎 1 𝑠 𝑠 1 superscript subscript 𝑘 1 𝑛 Bernoulli-number-B 2 𝑘 2 𝑘 Pochhammer 𝑠 2 𝑘 1 superscript 𝑎 1 𝑠 2 𝑘 1 Euler-Gamma 𝑠 superscript subscript 0 1 superscript 𝑒 𝑥 1 1 𝑥 1 2 superscript subscript 𝑘 1 𝑛 Bernoulli-number-B 2 𝑘 2 𝑘 superscript 𝑥 2 𝑘 1 superscript 𝑥 𝑠 1 superscript 𝑒 𝑎 𝑥 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1% -s}}{s-1}+\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}+% \frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1% }{x}+\frac{1}{2}-\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}e^{-% ax}\mathrm{d}x}}
\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}		 ℜ ⁡ s > - ( 2 ⁢ n + 1 ) , s ≠ 1 , ℜ ⁡ a > 0 , ℜ ⁡ s > 0 formulae-sequence 𝑠 2 𝑛 1 formulae-sequence 𝑠 1 formulae-sequence 𝑎 0 𝑠 0 {\displaystyle{\displaystyle\Re s>-(2n+1),s\neq 1,\Re a>0,\Re s>0}} 
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)

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]
Error Aborted - Skipped - Because timed out
25.11.E29 ζ ⁡ ( s , a ) = 1 2 ⁢ a - s + a 1 - s s - 1 + 2 ⁢ ∫ 0 ∞ sin ⁡ ( s ⁢ arctan ⁡ ( x / a ) ) ( a 2 + x 2 ) s / 2 ⁢ ( e 2 ⁢ π ⁢ x - 1 ) ⁢ d x Hurwitz-zeta 𝑠 𝑎 1 2 superscript 𝑎 𝑠 superscript 𝑎 1 𝑠 𝑠 1 2 superscript subscript 0 𝑠 𝑥 𝑎 superscript superscript 𝑎 2 superscript 𝑥 2 𝑠 2 superscript 𝑒 2 𝜋 𝑥 1 𝑥 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1% -s}}{s-1}+2\int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}\left(x/a% \right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\mathrm{d}x}}
\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}		 s ≠ 1 , ℜ ⁡ a > 0 formulae-sequence 𝑠 1 𝑎 0 {\displaystyle{\displaystyle s\neq 1,\Re a>0}} 
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)

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]
Failure Aborted Skipped - Because timed out Skipped - Because timed out
25.11.E30 ζ ⁡ ( s , a ) = Γ ⁡ ( 1 - s ) 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e a ⁢ z ⁢ z s - 1 1 - e z ⁢ d z Hurwitz-zeta 𝑠 𝑎 Euler-Gamma 1 𝑠 2 𝜋 𝑖 superscript subscript limit-from 0 superscript 𝑒 𝑎 𝑧 superscript 𝑧 𝑠 1 1 superscript 𝑒 𝑧 𝑧 {\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{\Gamma\left(1-s\right)% }{2\pi i}\int_{-\infty}^{(0+)}\frac{e^{az}z^{s-1}}{1-e^{z}}\mathrm{d}z}}
\Hurwitzzeta@{s}{a} = \frac{\EulerGamma@{1-s}}{2\pi i}\int_{-\infty}^{(0+)}\frac{e^{az}z^{s-1}}{1-e^{z}}\diff{z}		 s ≠ 1 , ℜ ⁡ a > 0 , ℜ ⁡ ( 1 - s ) > 0 formulae-sequence 𝑠 1 formulae-sequence 𝑎 0 1 𝑠 0 {\displaystyle{\displaystyle s\neq 1,\Re a>0,\Re(1-s)>0}} 
Zeta(0, s, a) = (GAMMA(1 - s))/(2*Pi*I)*int((exp(a*z)*(z)^(s - 1))/(1 - exp(z)), z = - infinity..(0 +))

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]
Error Failure - Error
25.11.E31 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 ⁢ e - a ⁢ x 2 ⁢ cosh ⁡ x ⁢ d x = 4 - s ⁢ ( ζ ⁡ ( s , 1 4 + 1 4 ⁢ a ) - ζ ⁡ ( s , 3 4 + 1 4 ⁢ a ) ) 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 superscript 𝑒 𝑎 𝑥 2 𝑥 𝑥 superscript 4 𝑠 Hurwitz-zeta 𝑠 1 4 1 4 𝑎 Hurwitz-zeta 𝑠 3 4 1 4 𝑎 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}% \frac{x^{s-1}e^{-ax}}{2\cosh x}\mathrm{d}x=4^{-s}\left(\zeta\left(s,\tfrac{1}{% 4}+\tfrac{1}{4}a\right)-\zeta\left(s,\tfrac{3}{4}+\tfrac{1}{4}a\right)\right)}}
\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)		 ℜ ⁡ s > 0 , ℜ ⁡ a > - 1 formulae-sequence 𝑠 0 𝑎 1 {\displaystyle{\displaystyle\Re s>0,\Re a>-1}} 
(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))

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])
Failure Successful Skipped - Because timed out Successful [Tested: 12]
25.11.E32 ∫ 0 a x n ⁢ ψ ⁡ ( x ) ⁢ d x = ( - 1 ) n - 1 ⁢ ζ ′ ⁡ ( - n ) + ( - 1 ) n ⁢ h ⁢ ( n ) ⁢ B n + 1 n + 1 - ∑ k = 0 n ( - 1 ) k ⁢ ( n k ) ⁢ h ⁢ ( k ) ⁢ B k + 1 ⁢ ( a ) k + 1 ⁢ a n - k + ∑ k = 0 n ( - 1 ) k ⁢ ( n k ) ⁢ ζ ′ ⁡ ( - k , a ) ⁢ a n - k superscript subscript 0 𝑎 superscript 𝑥 𝑛 digamma 𝑥 𝑥 superscript 1 𝑛 1 diffop Riemann-zeta 1 𝑛 superscript 1 𝑛 ℎ 𝑛 Bernoulli-number-B 𝑛 1 𝑛 1 superscript subscript 𝑘 0 𝑛 superscript 1 𝑘 binomial 𝑛 𝑘 ℎ 𝑘 Bernoulli-number-B 𝑘 1 𝑎 𝑘 1 superscript 𝑎 𝑛 𝑘 superscript subscript 𝑘 0 𝑛 superscript 1 𝑘 binomial 𝑛 𝑘 diffop Hurwitz-zeta 1 𝑘 𝑎 superscript 𝑎 𝑛 𝑘 {\displaystyle{\displaystyle\int_{0}^{a}x^{n}\psi\left(x\right)\mathrm{d}x=(-1% )^{n-1}\zeta'\left(-n\right)+(-1)^{n}h(n)\frac{B_{n+1}}{n+1}-\sum_{k=0}^{n}(-1% )^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}h(k)\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}% ^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}\zeta'\left(-k,a\right)a^{n-k}}}
\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}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
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)

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]
Failure Aborted
Failed [30 / 30]
Result: .9441788834-.4156250000*I
Test Values: {a = 3/2, h = 1/2*3^(1/2)+1/2*I, n = 3}


Result: 2.079687501-.7198836171*I
Test Values: {a = 3/2, h = -1/2+1/2*I*3^(1/2), n = 3}

... skip entries to safe data
 Skipped - Because timed out
25.11.E33 h ⁢ ( n ) = ∑ k = 1 n k - 1 ℎ 𝑛 superscript subscript 𝑘 1 𝑛 superscript 𝑘 1 {\displaystyle{\displaystyle h(n)=\sum_{k=1}^{n}k^{-1}}}
h(n) = \sum_{k=1}^{n}k^{-1}		 

h(n) = sum((k)^(- 1), k = 1..n)
 
h[n] == Sum[(k)^(- 1), {k, 1, n}, GenerateConditions->None]
 Skipped - no semantic math Skipped - no semantic math - -
25.11.E34 n ⁢ ∫ 0 a ζ ′ ⁡ ( 1 - n , x ) ⁢ d x = ζ ′ ⁡ ( - n , a ) - ζ ′ ⁡ ( - n ) + B n + 1 - B n + 1 ⁡ ( a ) n ⁢ ( n + 1 ) 𝑛 superscript subscript 0 𝑎 diffop Hurwitz-zeta 1 1 𝑛 𝑥 𝑥 diffop Hurwitz-zeta 1 𝑛 𝑎 diffop Riemann-zeta 1 𝑛 Bernoulli-number-B 𝑛 1 Bernoulli-polynomial-B 𝑛 1 𝑎 𝑛 𝑛 1 {\displaystyle{\displaystyle n\int_{0}^{a}\zeta'\left(1-n,x\right)\mathrm{d}x=% \zeta'\left(-n,a\right)-\zeta'\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a% \right)}{n(n+1)}}}
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)}		 ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle\Re a>0}} 
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))
 
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)]
 Failure Failure Manual Skip! Successful [Tested: 3]
25.11.E35 ∑ n = 0 ∞ ( - 1 ) n ( n + a ) s = 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 ⁢ e - a ⁢ x 1 + e - x ⁢ d x superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑛 𝑎 𝑠 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 superscript 𝑒 𝑎 𝑥 1 superscript 𝑒 𝑥 𝑥 {\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=% \frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}% \mathrm{d}x}}
\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}		 ℜ ⁡ a > 0 , ℜ ⁡ s > 0 , ℜ ⁡ a = 0 , ℑ ⁡ a ≠ 0 , 0 < ℜ ⁡ s , ℜ ⁡ s < 1 formulae-sequence 𝑎 0 formulae-sequence 𝑠 0 formulae-sequence 𝑎 0 formulae-sequence 𝑎 0 formulae-sequence 0 𝑠 𝑠 1 {\displaystyle{\displaystyle\Re a>0,\Re s>0,\Re a=0,\Im a\neq 0,0<\Re s,\Re s<% 1}} 
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)
 
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]
 Error Successful - -
25.11.E35 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 ⁢ e - a ⁢ x 1 + e - x ⁢ d x = 2 - s ⁢ ( ζ ⁡ ( s , 1 2 ⁢ a ) - ζ ⁡ ( s , 1 2 ⁢ ( 1 + a ) ) ) 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 superscript 𝑒 𝑎 𝑥 1 superscript 𝑒 𝑥 𝑥 superscript 2 𝑠 Hurwitz-zeta 𝑠 1 2 𝑎 Hurwitz-zeta 𝑠 1 2 1 𝑎 {\displaystyle{\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}% \frac{x^{s-1}e^{-ax}}{1+e^{-x}}\mathrm{d}x=2^{-s}\left(\zeta\left(s,\tfrac{1}{% 2}a\right)-\zeta\left(s,\tfrac{1}{2}(1+a)\right)\right)}}
\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)		 ℜ ⁡ a > 0 , ℜ ⁡ s > 0 , ℜ ⁡ a = 0 , ℑ ⁡ a ≠ 0 , 0 < ℜ ⁡ s , ℜ ⁡ s < 1 formulae-sequence 𝑎 0 formulae-sequence 𝑠 0 formulae-sequence 𝑎 0 formulae-sequence 𝑎 0 formulae-sequence 0 𝑠 𝑠 1 {\displaystyle{\displaystyle\Re a>0,\Re s>0,\Re a=0,\Im a\neq 0,0<\Re s,\Re s<% 1}} 
(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)))
 
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)])
 Error Successful - -
25.11.E36 ∑ n = 1 ∞ χ ⁢ ( n ) n s = k - s ⁢ ∑ r = 1 k - 1 χ ⁢ ( r ) ⁢ ζ ⁡ ( s , r k ) superscript subscript 𝑛 1 𝜒 𝑛 superscript 𝑛 𝑠 superscript 𝑘 𝑠 superscript subscript 𝑟 1 𝑘 1 𝜒 𝑟 Hurwitz-zeta 𝑠 𝑟 𝑘 {\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}=k^{-s}% \sum_{r=1}^{k-1}\chi(r)\zeta\left(s,\frac{r}{k}\right)}}
\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}} = k^{-s}\sum_{r=1}^{k-1}\chi(r)\Hurwitzzeta@{s}{\frac{r}{k}}		 ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle\Re s>1}} 
sum((chi(n))/((n)^(s)), n = 1..infinity) = (k)^(- s)* sum(chi(r)* Zeta(0, s, (r)/(k)), r = 1..k - 1)
 
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]
 Failure Failure
Failed [60 / 60]
Result: Float(infinity)+Float(infinity)*I
Test Values: {chi = 1/2*3^(1/2)+1/2*I, s = 3/2, k = 1}

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

... skip entries to safe data
Failed [60 / 60]
Result: Complex[-1.264704103160249, -0.7301772544047939]
Test Values: {Rule[a, 1], Rule[k, 1], Rule[s, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[-2.727214191729021, -1.574557847732518]
Test Values: {Rule[a, 1], Rule[k, 2], Rule[s, 1.5], Rule[χ, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

... skip entries to safe data
25.11.E37 ∑ k = 1 ∞ ( - 1 ) k k ⁢ ζ ⁡ ( n ⁢ k , a ) = - n ⁢ ln ⁡ Γ ⁡ ( a ) + ln ⁡ ( ∏ j = 0 n - 1 Γ ⁡ ( a - e ( 2 ⁢ j + 1 ) ⁢ π ⁢ i / n ) ) superscript subscript 𝑘 1 superscript 1 𝑘 𝑘 Hurwitz-zeta 𝑛 𝑘 𝑎 𝑛 Euler-Gamma 𝑎 superscript subscript product 𝑗 0 𝑛 1 Euler-Gamma 𝑎 superscript 𝑒 2 𝑗 1 𝜋 𝑖 𝑛 {\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\zeta\left(nk% ,a\right)=-n\ln\Gamma\left(a\right)+\ln\left(\prod_{j=0}^{n-1}\Gamma\left(a-e^% {(2j+1)\pi i/n}\right)\right)}}
\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}}}		 ℜ ⁡ a ≥ 1 𝑎 1 {\displaystyle{\displaystyle\Re a\geq 1}} 
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))
 
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]]
 Failure Failure Successful [Tested: 2]
Failed [1 / 3]
Result: NSum[Times[Power[-1, k], Power[k, -1], Zeta[k]]
Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]], {Rule[a, 1], Rule[n, 1]}

25.11.E38 ∑ k = 1 ∞ ( n + k k ) ⁢ ζ ⁡ ( n + k + 1 , a ) ⁢ z k = ( - 1 ) n n ! ⁢ ( ψ ( n ) ⁡ ( a ) - ψ ( n ) ⁡ ( a - z ) ) superscript subscript 𝑘 1 binomial 𝑛 𝑘 𝑘 Hurwitz-zeta 𝑛 𝑘 1 𝑎 superscript 𝑧 𝑘 superscript 1 𝑛 𝑛 digamma 𝑛 𝑎 digamma 𝑛 𝑎 𝑧 {\displaystyle{\displaystyle\sum_{k=1}^{\infty}\genfrac{(}{)}{0.0pt}{}{n+k}{k}% \zeta\left(n+k+1,a\right)z^{k}=\frac{(-1)^{n}}{n!}\left({\psi^{(n)}}\left(a% \right)-{\psi^{(n)}}\left(a-z\right)\right)}}
\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)		 ℜ ⁡ a > 0 , | z | < | a | formulae-sequence 𝑎 0 𝑧 𝑎 {\displaystyle{\displaystyle\Re a>0,|z|<|a|}} 
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) ) ))
 
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))
 Failure Failure Manual Skip! Skipped - Because timed out
25.11.E39 ∑ k = 2 ∞ k 2 k ⁢ ζ ⁡ ( k + 1 , 3 4 ) = 8 ⁢ G superscript subscript 𝑘 2 𝑘 superscript 2 𝑘 Hurwitz-zeta 𝑘 1 3 4 8 𝐺 {\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{k}{2^{k}}\zeta\left(k+1,% \tfrac{3}{4}\right)=8G}}
\sum_{k=2}^{\infty}\frac{k}{2^{k}}\Hurwitzzeta@{k+1}{\tfrac{3}{4}} = 8G		 

sum((k)/((2)^(k))*Zeta(0, k + 1, (3)/(4)), k = 2..infinity) = 8*G
 
Sum[Divide[k,(2)^(k)]*HurwitzZeta[k + 1, Divide[3,4]], {k, 2, Infinity}, GenerateConditions->None] == 8*G
 Failure Failure
Failed [10 / 10]
Result: .399521521-4.000000000*I
Test Values: {G = 1/2*3^(1/2)+1/2*I}

Result: 11.32772475-6.928203232*I
Test Values: {G = -1/2+1/2*I*3^(1/2)}

... skip entries to safe data
Failed [10 / 10]
Result: Complex[0.39952152314224243, -3.9999999999999996]
Test Values: {Rule[a, 1], Rule[G, Power[E, Times[Complex[0, Rational[1, 6]], Pi]]]}

Result: Complex[11.327724753417751, -6.92820323027551]
Test Values: {Rule[a, 1], Rule[G, Power[E, Times[Complex[0, Rational[2, 3]], Pi]]]}

... skip entries to safe data
25.11.E40 G ≡ ∑ n = 0 ∞ ( - 1 ) n ( 2 ⁢ n + 1 ) 2 = 0.91596 55941 772 ⁢ … equal-by definition 𝐺 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 2 𝑛 1 2 0.91596 55941 772 … {\displaystyle{\displaystyle G\equiv\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^% {2}}=0.91596\;55941\;772\dots}}
G\defeq\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}} = 0.91596\;55941\;772\dots		 

G = sum(((- 1)^(n))/((2*n + 1)^(2)), n = 0..infinity) = 0.9159655941772
 
G == Sum[Divide[(- 1)^(n),(2*n + 1)^(2)], {n, 0, Infinity}, GenerateConditions->None] == 0.9159655941772
 Failure Skipped - Invalid test case: dots Error -
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