Zeta and Related Functions - 25.11 Hurwitz Zeta Function

DLMF	Formula	Constraints	Maple	Mathematica	Symbolic Maple	Symbolic Mathematica	Numeric Maple	Numeric Mathematica
25.11.E1	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}}}$ \Hurwitzzeta@{s}{a} = \sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}	$$	Zeta(0, s, a) = sum((1)/((n + a)^(s)), n = 0..infinity)	HurwitzZeta[s, a] == Sum[Divide[1,(n + a)^(s)], {n, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 2]	Successful [Tested: 2]
25.11.E2	${\displaystyle{\displaystyle\zeta\left(s,1\right)=\zeta\left(s\right)}}$ \Hurwitzzeta@{s}{1} = \Riemannzeta@{s}	${\displaystyle }$	Zeta(0, s, 1) = Zeta(s)	HurwitzZeta[s, 1] == Zeta[s]	Successful	Successful	-	Successful [Tested: 6]
25.11.E3	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\zeta\left(s,a+1\right)+a^{-s}}}$ \Hurwitzzeta@{s}{a} = \Hurwitzzeta@{s}{a+1}+a^{-s}	${\displaystyle }$	Zeta(0, s, a) = Zeta(0, s, a + 1)+ (a)^(- s)	HurwitzZeta[s, a] == HurwitzZeta[s, a + 1]+ (a)^(- s)	Failure	Successful	Error	Failed [3 / 36] Result: Indeterminate Test Values: {Rule[a, -2], Rule[s, 1.5]} Result: Indeterminate Test Values: {Rule[a, -2], Rule[s, 0.5]} ... skip entries to safe data
25.11.E4	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\zeta\left(s,a+m\right)+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{s}}}}$ \Hurwitzzeta@{s}{a} = \Hurwitzzeta@{s}{a+m}+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{s}}	${\displaystyle }$	Zeta(0, s, a) = Zeta(0, s, a + m)+ sum((1)/((n + a)^(s)), n = 0..m - 1)	HurwitzZeta[s, a] == HurwitzZeta[s, a + m]+ Sum[Divide[1,(n + a)^(s)], {n, 0, m - 1}, GenerateConditions->None]	Failure	Successful	Error	Successful [Tested: 36]
25.11.E5	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{N}\frac{1}{(n+a)^{s}}+\frac{(N+a)^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{(x+a)^{s+1}}\mathrm{d}x}}$ \Hurwitzzeta@{s}{a} = \sum_{n=0}^{N}\frac{1}{(n+a)^{s}}+\frac{(N+a)^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\floor{x}}{(x+a)^{s+1}}\diff{x}	$$	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	${\displaystyle{\displaystyle\zeta\left(s,\tfrac{1}{2}a\right)=\zeta\left(s,\tfrac{1}{2}a+\tfrac{1}{2}\right)+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}}}$ \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}}	$$	Zeta(0, s, (1)/(2)*a) = Zeta(0, s, (1)/(2)*a +(1)/(2))+ (2)^(s)* sum(((- 1)^(n))/((n + a)^(s)), n = 0..infinity)	HurwitzZeta[s, Divide[1,2]*a] == HurwitzZeta[s, Divide[1,2]*a +Divide[1,2]]+ (2)^(s)* Sum[Divide[(- 1)^(n),(n + a)^(s)], {n, 0, Infinity}, GenerateConditions->None]	Failure	Successful	Successful [Tested: 3]	Successful [Tested: 3]
25.11.E9	${\displaystyle{\displaystyle\zeta\left(1-s,a\right)=\frac{2\Gamma\left(s\right)}{(2\pi)^{s}}\*\sum_{n=1}^{\infty}\frac{1}{n^{s}}\cos\left(\tfrac{1}{2}\pi s-2n\pi a\right)}}$ \Hurwitzzeta@{1-s}{a} = \frac{2\EulerGamma@{s}}{(2\pi)^{s}}\*\sum_{n=1}^{\infty}\frac{1}{n^{s}}\cos@{\tfrac{1}{2}\pi s-2n\pi a}	$$	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	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{{\left(s\right)_{n}}}{n!}\zeta\left(n+s\right)(1-a)^{n}}}$ \Hurwitzzeta@{s}{a} = \sum_{n=0}^{\infty}\frac{\Pochhammersym{s}{n}}{n!}\Riemannzeta@{n+s}(1-a)^{n}	$s\ne 1,|a-1|<1$	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	${\displaystyle{\displaystyle\zeta\left(s,\tfrac{1}{2}\right)=(2^{s}-1)\zeta\left(s\right)}}$ \Hurwitzzeta@{s}{\tfrac{1}{2}} = (2^{s}-1)\Riemannzeta@{s}	$s\ne 1$	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	${\displaystyle{\displaystyle\zeta\left(n+1,a\right)=\frac{(-1)^{n+1}{\psi^{(n)}}\left(a\right)}{n!}}}$ \Hurwitzzeta@{n+1}{a} = \frac{(-1)^{n+1}\digamma^{(n)}@{a}}{n!}	${\displaystyle }$	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	${\displaystyle{\displaystyle\zeta\left(0,a\right)=\tfrac{1}{2}-a}}$ \Hurwitzzeta@{0}{a} = \tfrac{1}{2}-a	${\displaystyle }$	Zeta(0, 0, a) = (1)/(2)- a	HurwitzZeta[0, a] == Divide[1,2]- a	Successful	Successful	-	Successful [Tested: 6]
25.11.E14	${\displaystyle{\displaystyle\zeta\left(-n,a\right)=-\frac{B_{n+1}\left(a\right)}{n+1}}}$ \Hurwitzzeta@{-n}{a} = -\frac{\BernoullipolyB{n+1}@{a}}{n+1}	${\displaystyle }$	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	${\displaystyle{\displaystyle\zeta\left(s,ka\right)=k^{-s}\*\sum_{n=0}^{k-1}\zeta\left(s,a+\frac{n}{k}\right)}}$ \Hurwitzzeta@{s}{ka} = k^{-s}\*\sum_{n=0}^{k-1}\Hurwitzzeta@{s}{a+\frac{n}{k}}	$s\ne 1$	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	${\displaystyle{\displaystyle\zeta\left(1-s,\frac{h}{k}\right)=\frac{2\Gamma\left(s\right)}{(2\pi k)^{s}}\*\sum_{r=1}^{k}\cos\left(\frac{\pi s}{2}-\frac{2\pi rh}{k}\right)\zeta\left(s,\frac{r}{k}\right)}}$ \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}}	$$	Zeta(0, 1 - s, (h)/(k)) = (2*GAMMA(s))/((2*Pi*k)^(s))* sum(cos((Pi*s)/(2)-(2*Pi*r*h)/(k))*Zeta(0, s, (r)/(k)), r = 1..k)	HurwitzZeta[1 - s, Divide[h,k]] == Divide[2*Gamma[s],(2*Pi*k)^(s)]* Sum[Cos[Divide[Pi*s,2]-Divide[2*Pi*r*h,k]]*HurwitzZeta[s, Divide[r,k]], {r, 1, k}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
25.11.E17	${\displaystyle{\displaystyle\frac{\partial}{\partial a}\zeta\left(s,a\right)=-s\zeta\left(s+1,a\right)}}$ \pderiv{}{a}\Hurwitzzeta@{s}{a} = -s\Hurwitzzeta@{s+1}{a}	$$	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	${\displaystyle{\displaystyle\zeta'\left(0,a\right)=\ln\Gamma\left(a\right)-\tfrac{1}{2}\ln\left(2\pi\right)}}$ \Hurwitzzeta'@{0}{a} = \ln@@{\EulerGamma@{a}}-\tfrac{1}{2}\ln@{2\pi}	$$	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	${\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}}	${\displaystyle }$	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	${\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}}	${\displaystyle }$	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	${\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}}	${\displaystyle }$	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	${\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\ne 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	${\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}	$$	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	${\displaystyle{\displaystyle\zeta\left(s,a\right)=-s\int_{-a}^{\infty}\frac{x-\left\lfloor x\right\rfloor-\frac{1}{2}}{(x+a)^{s+1}}\mathrm{d}x}}$ \Hurwitzzeta@{s}{a} = -s\int_{-a}^{\infty}\frac{x-\floor{x}-\frac{1}{2}}{(x+a)^{s+1}}\diff{x}	$$	Zeta(0, s, a) = - s*int((x - floor(x)-(1)/(2))/((x + a)^(s + 1)), x = - a..infinity)	HurwitzZeta[s, a] == - s*Integrate[Divide[x - Floor[x]-Divide[1,2],(x + a)^(s + 1)], {x, - a, Infinity}, GenerateConditions->None]	Error	Aborted	-	Skip - No test values generated
25.11.E27	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{ax}}\mathrm{d}x}}$ \Hurwitzzeta@{s}{a} = \frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{ax}}\diff{x}	$$	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	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}e^{-ax}\mathrm{d}x}}$ \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}	$$	Zeta(0, s, a) = (1)/(2)*(a)^(- s)+((a)^(1 - s))/(s - 1)+ sum((bernoulli(2*k))/(factorial(2*k))*pochhammer(s, 2*k - 1)*(a)^(1 - s - 2*k)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2)- sum((bernoulli(2*k))/(factorial(2*k))*(x)^(2*k - 1), k = 1..n))*(x)^(s - 1)* exp(- a*x), x = 0..infinity), k = 1..n)	HurwitzZeta[s, a] == Divide[1,2]*(a)^(- s)+Divide[(a)^(1 - s),s - 1]+ Sum[Divide[BernoulliB[2*k],(2*k)!]*Pochhammer[s, 2*k - 1]*(a)^(1 - s - 2*k)+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2]- Sum[Divide[BernoulliB[2*k],(2*k)!]*(x)^(2*k - 1), {k, 1, n}, GenerateConditions->None])*(x)^(s - 1)* Exp[- a*x], {x, 0, Infinity}, GenerateConditions->None], {k, 1, n}, GenerateConditions->None]	Error	Aborted	-	Skipped - Because timed out
25.11.E29	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+2\int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\mathrm{d}x}}$ \Hurwitzzeta@{s}{a} = \frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+2\int_{0}^{\infty}\frac{\sin@{s\atan@{x/a}}}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1)}\diff{x}	$$	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	${\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{\Gamma\left(1-s\right)}{2\pi i}\int_{-\infty}^{(0+)}\frac{e^{az}z^{s-1}}{1-e^{z}}\mathrm{d}z}}$ \Hurwitzzeta@{s}{a} = \frac{\EulerGamma@{1-s}}{2\pi i}\int_{-\infty}^{(0+)}\frac{e^{az}z^{s-1}}{1-e^{z}}\diff{z}	$$	Zeta(0, s, a) = (GAMMA(1 - s))/(2*Pi*I)*int((exp(a*z)*(z)^(s - 1))/(1 - exp(z)), z = - infinity..(0 +))	HurwitzZeta[s, a] == Divide[Gamma[1 - s],2*Pi*I]*Integrate[Divide[Exp[a*z]*(z)^(s - 1),1 - Exp[z]], {z, - Infinity, (0 +)}, GenerateConditions->None]	Error	Failure	-	Error
25.11.E31	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{2\cosh x}\mathrm{d}x=4^{-s}\left(\zeta\left(s,\tfrac{1}{4}+\tfrac{1}{4}a\right)-\zeta\left(s,\tfrac{3}{4}+\tfrac{1}{4}a\right)\right)}}$ \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{2\cosh@@{x}}\diff{x} = 4^{-s}\left(\Hurwitzzeta@{s}{\tfrac{1}{4}+\tfrac{1}{4}a}-\Hurwitzzeta@{s}{\tfrac{3}{4}+\tfrac{1}{4}a}\right)	$$	(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	${\displaystyle{\displaystyle\int_{0}^{a}x^{n}\psi\left(x\right)\mathrm{d}x=(-1)^{n-1}\zeta'\left(-n\right)+(-1)^{n}h(n)\frac{B_{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}h(k)\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}\zeta'\left(-k,a\right)a^{n-k}}}$ \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}	$$	int((x)^(n)* Psi(x), x = 0..a) = (- 1)^(n - 1)* subs( temp=- n, diff( Zeta(temp), temp$(1) ) )+(- 1)^(n)* h(n)*(bernoulli(n + 1))/(n + 1)- sum((- 1)^(k)*binomial(n,k)*h(k)*(bernoulli(k + 1)*(a))/(k + 1)*(a)^(n - k), k = 0..n)+ sum((- 1)^(k)*binomial(n,k)*subs( temp=- k, diff( Zeta(0, temp, a), temp$(1) ) )*(a)^(n - k), k = 0..n)	Integrate[(x)^(n)* PolyGamma[x], {x, 0, a}, GenerateConditions->None] == (- 1)^(n - 1)* (D[Zeta[temp], {temp, 1}]/.temp-> - n)+(- 1)^(n)* h[n]*Divide[BernoulliB[n + 1],n + 1]- Sum[(- 1)^(k)*Binomial[n,k]*h[k]*Divide[BernoulliB[k + 1]*(a),k + 1]*(a)^(n - k), {k, 0, n}, GenerateConditions->None]+ Sum[(- 1)^(k)*Binomial[n,k]*(D[HurwitzZeta[temp, a], {temp, 1}]/.temp-> - k)*(a)^(n - k), {k, 0, n}, GenerateConditions->None]	Failure	Aborted	Failed [30 / 30] Result: .9441788834-.4156250000*I Test Values: {a = 3/2, h = 1/2*3^(1/2)+1/2*I, n = 3} Result: 2.079687501-.7198836171*I Test Values: {a = 3/2, h = -1/2+1/2*I*3^(1/2), n = 3} ... skip entries to safe data	Skipped - Because timed out
25.11.E33	${\displaystyle{\displaystyle h(n)=\sum_{k=1}^{n}k^{-1}}}$ h(n) = \sum_{k=1}^{n}k^{-1}	${\displaystyle }$	h(n) = sum((k)^(- 1), k = 1..n)	h[n] == Sum[(k)^(- 1), {k, 1, n}, GenerateConditions->None]	Skipped - no semantic math	Skipped - no semantic math	-	-
25.11.E34	${\displaystyle{\displaystyle n\int_{0}^{a}\zeta'\left(1-n,x\right)\mathrm{d}x=\zeta'\left(-n,a\right)-\zeta'\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a\right)}{n(n+1)}}}$ n\int_{0}^{a}\Hurwitzzeta'@{1-n}{x}\diff{x} = \Hurwitzzeta'@{-n}{a}-\Riemannzeta'@{-n}+\frac{\BernoullinumberB{n+1}-\BernoullipolyB{n+1}@{a}}{n(n+1)}	$$	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	${\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\mathrm{d}x}}$ \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}	$$	sum(((- 1)^(n))/((n + a)^(s)), n = 0..infinity) = (1)/(GAMMA(s))*int(((x)^(s - 1)* exp(- a*x))/(1 + exp(- x)), x = 0..infinity)	Sum[Divide[(- 1)^(n),(n + a)^(s)], {n, 0, Infinity}, GenerateConditions->None] == Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1)* Exp[- a*x],1 + Exp[- x]], {x, 0, Infinity}, GenerateConditions->None]	Error	Successful	-	-
25.11.E35	${\displaystyle{\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\mathrm{d}x=2^{-s}\left(\zeta\left(s,\tfrac{1}{2}a\right)-\zeta\left(s,\tfrac{1}{2}(1+a)\right)\right)}}$ \frac{1}{\EulerGamma@{s}}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\diff{x} = 2^{-s}\left(\Hurwitzzeta@{s}{\tfrac{1}{2}a}-\Hurwitzzeta@{s}{\tfrac{1}{2}(1+a)}\right)	$$	(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	${\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}=k^{-s}\sum_{r=1}^{k-1}\chi(r)\zeta\left(s,\frac{r}{k}\right)}}$ \sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}} = k^{-s}\sum_{r=1}^{k-1}\chi(r)\Hurwitzzeta@{s}{\frac{r}{k}}	$$	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	${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\zeta\left(nk,a\right)=-n\ln\Gamma\left(a\right)+\ln\left(\prod_{j=0}^{n-1}\Gamma\left(a-e^{(2j+1)\pi i/n}\right)\right)}}$ \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}}}	$$	sum(((- 1)^(k))/(k)*Zeta(0, n*k, a), k = 1..infinity) = - n*ln(GAMMA(a))+ ln(product(GAMMA(a - exp((2*j + 1)*Pi*I/n)), j = 0..n - 1))	Sum[Divide[(- 1)^(k),k]*HurwitzZeta[n*k, a], {k, 1, Infinity}, GenerateConditions->None] == - n*Log[Gamma[a]]+ Log[Product[Gamma[a - Exp[(2*j + 1)*Pi*I/n]], {j, 0, n - 1}, GenerateConditions->None]]	Failure	Failure	Successful [Tested: 2]	Failed [1 / 3] Result: NSum[Times[Power[-1, k], Power[k, -1], Zeta[k]] Test Values: {k, 1, DirectedInfinity[1]}, Rule[GenerateConditions, None]], {Rule[a, 1], Rule[n, 1]}
25.11.E38	${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\genfrac{(}{)}{0.0pt}{}{n+k}{k}\zeta\left(n+k+1,a\right)z^{k}=\frac{(-1)^{n}}{n!}\left({\psi^{(n)}}\left(a\right)-{\psi^{(n)}}\left(a-z\right)\right)}}$ \sum_{k=1}^{\infty}\binom{n+k}{k}\Hurwitzzeta@{n+k+1}{a}z^{k} = \frac{(-1)^{n}}{n!}\left(\digamma^{(n)}@{a}-\digamma^{(n)}@{a-z}\right)	$$	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	${\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{k}{2^{k}}\zeta\left(k+1,\tfrac{3}{4}\right)=8G}}$ \sum_{k=2}^{\infty}\frac{k}{2^{k}}\Hurwitzzeta@{k+1}{\tfrac{3}{4}} = 8G	${\displaystyle }$	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	${\displaystyle{\displaystyle G\equiv\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}=0.91596\;55941\;772\dots}}$ G\defeq\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}} = 0.91596\;55941\;772\dots	${\displaystyle }$	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	-