Hurwitz Zeta Function

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Hurwitz Zeta Function

Definition

\HurwitzZeta ⁢ @ ⁢ s ⁢ a = ∑ n = 0 ∞ 1 ( n + a ) s \HurwitzZeta @ 𝑠 𝑎 superscript subscript 𝑛 0 1 superscript 𝑛 𝑎 𝑠 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\sum_{n=0}^{% \infty}\frac{1}{(n+a)^{s}}}}} {\displaystyle \HurwitzZeta@{s}{a} = \sum_{n=0}^\infty \frac{1}{(n+a)^s} }

Constraint(s): ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}} &
a ≠ 0 , - 1 , - 2 , … 𝑎 0 1 2 … {\displaystyle{\displaystyle{\displaystyle a\neq 0,-1,-2,\dots}}}


\HurwitzZeta ⁢ @ ⁢ s ⁢ 1 = \RiemannZeta ⁢ @ ⁢ s \HurwitzZeta @ 𝑠 1 \RiemannZeta @ 𝑠 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{1}=\RiemannZeta@{s}% }}} {\displaystyle \HurwitzZeta@{s}{1} = \RiemannZeta@{s} }
\HurwitzZeta ⁢ @ ⁢ s ⁢ a = \HurwitzZeta ⁢ @ ⁢ s ⁢ a + 1 + a - s \HurwitzZeta @ 𝑠 𝑎 \HurwitzZeta @ 𝑠 𝑎 1 superscript 𝑎 𝑠 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\HurwitzZeta@{s}% {a+1}+a^{-s}}}} {\displaystyle \HurwitzZeta@{s}{a} = \HurwitzZeta@{s}{a+1} + a^{-s} }
\HurwitzZeta ⁢ @ ⁢ s ⁢ a = \HurwitzZeta ⁢ @ ⁢ s ⁢ a + m + ∑ n = 0 m - 1 1 ( n + a ) s \HurwitzZeta @ 𝑠 𝑎 \HurwitzZeta @ 𝑠 𝑎 𝑚 superscript subscript 𝑛 0 𝑚 1 1 superscript 𝑛 𝑎 𝑠 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\HurwitzZeta@{s}% {a+m}+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{s}}}}} {\displaystyle \HurwitzZeta@{s}{a} = \HurwitzZeta@{s}{a+m} + \sum_{n=0}^{m-1} \frac{1}{(n+a)^s} }

Constraint(s): m = 1 , 2 , 3 , … 𝑚 1 2 3 … {\displaystyle{\displaystyle{\displaystyle m=1,2,3,\dots}}}


Representations by the Euler-Maclaurin Formula

\HurwitzZeta ⁢ @ ⁢ 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 \HurwitzZeta @ 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{\displaystyle\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-\left% \lfloor x\right\rfloor}{(x+a)^{s+1}}\mathrm{d}x}}} {\displaystyle \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} }

Constraint(s): s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}} &
a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a>0}}} &
N = 0 , 1 , 2 , 3 , … 𝑁 0 1 2 3 … {\displaystyle{\displaystyle{\displaystyle N=0,1,2,3,\dots}}}


\HurwitzZeta ⁢ @ ⁢ s ⁢ a = 1 a s ⁢ ( 1 2 + a s - 1 ) - s ⁢ ( s + 1 ) ⁢ ∫ 0 ∞ \PeriodicBernoulliB ⁢ 2 ⁢ @ ⁢ x ( x + a ) s + 2 ⁢ d x \HurwitzZeta @ 𝑠 𝑎 1 superscript 𝑎 𝑠 1 2 𝑎 𝑠 1 𝑠 𝑠 1 superscript subscript 0 \PeriodicBernoulliB 2 @ 𝑥 superscript 𝑥 𝑎 𝑠 2 𝑥 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{1}{a^{s}}% \left(\frac{1}{2}+\frac{a}{s-1}\right)-s(s+1)\int_{0}^{\infty}\frac{% \PeriodicBernoulliB{2}@{x}}{(x+a)^{s+2}}\mathrm{d}x}}} {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{a^s} \left( \frac{1}{2} + \frac{a}{s-1} \right) - s (s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x}}{(x+a)^{s+2}} \diff{x} }

Constraint(s): s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
ℜ ⁡ s > - 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>-1}}} &
a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a>0}}}


\HurwitzZeta ⁢ @ ⁢ s ⁢ a = 1 a s + 1 ( 1 + a ) s ⁢ ( 1 2 + 1 + a s - 1 ) + ∑ k = 1 n ( s + 2 ⁢ k - 2 2 ⁢ k - 1 ) ⁢ \BernoulliB ⁢ 2 ⁢ k 2 ⁢ k ⁢ 1 ( 1 + a ) s + 2 ⁢ k - 1 - ( s + 2 ⁢ n 2 ⁢ n + 1 ) ⁢ ∫ 1 ∞ \PeriodicBernoulliB ⁢ 2 ⁢ n + 1 ⁢ @ ⁢ x ( x + a ) s + 2 ⁢ n + 1 ⁢ d x \HurwitzZeta @ 𝑠 𝑎 1 superscript 𝑎 𝑠 1 superscript 1 𝑎 𝑠 1 2 1 𝑎 𝑠 1 superscript subscript 𝑘 1 𝑛 binomial 𝑠 2 𝑘 2 2 𝑘 1 \BernoulliB 2 𝑘 2 𝑘 1 superscript 1 𝑎 𝑠 2 𝑘 1 binomial 𝑠 2 𝑛 2 𝑛 1 superscript subscript 1 \PeriodicBernoulliB 2 𝑛 1 @ 𝑥 superscript 𝑥 𝑎 𝑠 2 𝑛 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{1}{a^{s}}+% \frac{1}{(1+a)^{s}}\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum_{k=1}^{n}% \genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}\frac{\BernoulliB{2k}}{2k}\frac{1}{(1+a)^% {s+2k-1}}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}\int_{1}^{\infty}\frac{% \PeriodicBernoulliB{2n+1}@{x}}{(x+a)^{s+2n+1}}\mathrm{d}x}}} {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{a^s} + \frac{1}{(1+a)^s} \left( \frac{1}{2} + \frac{1+a}{s-1} \right) + \sum_{k=1}^n \binom{s+2k-2}{2k-1} \frac{\BernoulliB{2k}}{2k} \frac{1}{(1+a)^{s+2k-1}} - \binom{s+2n}{2n+1} \int_1^\infty \frac{\PeriodicBernoulliB{2n+1}@{x}}{(x+a)^{s+2n+1}} \diff{x} }

Constraint(s): s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a>0}}} &
n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}} &
ℜ ⁡ s > - 2 ⁢ n 𝑠 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\Re{s}>-2n}}}


Series Representations

\HurwitzZeta ⁢ @ ⁢ s ⁢ 1 2 ⁢ a = \HurwitzZeta ⁢ @ ⁢ s ⁢ 1 2 ⁢ a + 1 2 + 2 s ⁢ ∑ n = 0 ∞ ( - 1 ) n ( n + a ) s \HurwitzZeta @ 𝑠 1 2 𝑎 \HurwitzZeta @ 𝑠 1 2 𝑎 1 2 superscript 2 𝑠 superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑛 𝑎 𝑠 {\displaystyle{\displaystyle{\displaystyle\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}}}}} {\displaystyle \HurwitzZeta@{s}{\tfrac{1}{2} a} = \HurwitzZeta@{s}{\tfrac{1}{2} a + \tfrac{1}{2}} + 2^s \sum_{n=0}^\infty \frac{\opminus^n}{(n+a)^s} }

Constraint(s): ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}} &
s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
0 < a ≤ 1 0 𝑎 1 {\displaystyle{\displaystyle{\displaystyle 0<a\leq 1}}}


\HurwitzZeta ⁢ @ ⁢ 1 - s ⁢ a = 2 ⁢ Γ ⁡ ( s ) ( 2 ⁢ π ) s ⁢ ∑ n = 1 ∞ 1 n s ⁢ cos ⁡ ( 1 2 ⁢ π ⁢ s - 2 ⁢ n ⁢ π ⁢ a ) \HurwitzZeta @ 1 𝑠 𝑎 2 Euler-Gamma 𝑠 superscript 2 𝑠 superscript subscript 𝑛 1 1 superscript 𝑛 𝑠 1 2 𝑠 2 𝑛 𝑎 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{1-s}{a}=\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)}}} {\displaystyle \HurwitzZeta@{1-s}{a} = \frac{2\EulerGamma@{s}}{(2\cpi)^s} \* \sum_{n=1}^\infty \frac{1}{n^s} \cos@{\tfrac{1}{2} \cpi s - 2n \cpi a} }

Constraint(s): ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}} &
0 < a ≤ 1 0 𝑎 1 {\displaystyle{\displaystyle{\displaystyle 0<a\leq 1}}}


\HurwitzZeta ⁢ @ ⁢ s ⁢ a = ∑ n = 0 ∞ Γ ⁡ ( n + s ) n ! ⁢ Γ ⁡ ( s ) ⁢ \RiemannZeta ⁢ @ ⁢ n + s ⁢ ( 1 - a ) n \HurwitzZeta @ 𝑠 𝑎 superscript subscript 𝑛 0 Euler-Gamma 𝑛 𝑠 𝑛 Euler-Gamma 𝑠 \RiemannZeta @ 𝑛 𝑠 superscript 1 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\sum_{n=0}^{% \infty}\frac{\Gamma\left(n+s\right)}{n!\Gamma\left(s\right)}\RiemannZeta@{n+s}% (1-a)^{n}}}} {\displaystyle \HurwitzZeta@{s}{a} = \sum_{n=0}^\infty \frac{\EulerGamma@{n+s}}{n! \EulerGamma@{s}} \RiemannZeta@{n+s} (1-a)^n }

Constraint(s): s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
| a - 1 | < 1 𝑎 1 1 {\displaystyle{\displaystyle{\displaystyle|a-1|<1}}}


Special Values

\HurwitzZeta ⁢ @ ⁢ s ⁢ 1 2 = ( 2 s - 1 ) ⁢ \RiemannZeta ⁢ @ ⁢ s \HurwitzZeta @ 𝑠 1 2 superscript 2 𝑠 1 \RiemannZeta @ 𝑠 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{\tfrac{1}{2}}=(2^{s% }-1)\RiemannZeta@{s}}}} {\displaystyle \HurwitzZeta@{s}{\tfrac{1}{2}} = (2^s - 1) \RiemannZeta@{s} }

Constraint(s): s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}}


\HurwitzZeta ⁢ @ ⁢ n + 1 ⁢ a = ( - 1 ) n + 1 ⁢ ψ ( n ) ⁡ ( a ) n ! \HurwitzZeta @ 𝑛 1 𝑎 superscript 1 𝑛 1 digamma 𝑛 𝑎 𝑛 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{n+1}{a}=\frac{(-1)^{n+% 1}{\psi^{(n)}}\left(a\right)}{n!}}}} {\displaystyle \HurwitzZeta@{n+1}{a} = \frac{\opminus^{n+1} \digamma^{(n)}@{a}}{n!} }

Constraint(s): n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


\HurwitzZeta ⁢ @ ⁢ 0 ⁢ a = 1 2 - a \HurwitzZeta @ 0 𝑎 1 2 𝑎 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{0}{a}=\tfrac{1}{2}-a}}} {\displaystyle \HurwitzZeta@{0}{a} = \tfrac{1}{2} - a }

Constraint(s): ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


\HurwitzZeta ⁢ @ - n ⁢ a = - \BernoulliB ⁢ n + 1 ⁢ @ ⁢ a n + 1 \HurwitzZeta @ 𝑛 𝑎 \BernoulliB 𝑛 1 @ 𝑎 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{-n}{a}=-\frac{% \BernoulliB{n+1}@{a}}{n+1}}}} {\displaystyle \HurwitzZeta@{-n}{a} = -\frac{\BernoulliB{n+1}@{a}}{n+1} }

Constraint(s): n = 0 , 1 , 2 , … 𝑛 0 1 2 … {\displaystyle{\displaystyle{\displaystyle n=0,1,2,\dots}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


\HurwitzZeta ⁢ @ ⁢ s ⁢ k ⁢ a = k - s ⁢ ∑ n = 0 k - 1 \HurwitzZeta ⁢ @ ⁢ s ⁢ a + n k \HurwitzZeta @ 𝑠 𝑘 𝑎 superscript 𝑘 𝑠 superscript subscript 𝑛 0 𝑘 1 \HurwitzZeta @ 𝑠 𝑎 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{ka}=k^{-s}\*\sum_{n% =0}^{k-1}\HurwitzZeta@{s}{a+\frac{n}{k}}}}} {\displaystyle \HurwitzZeta@{s}{ka} = k^{-s} \* \sum_{n=0}^{k-1} \HurwitzZeta@{s}{a+\frac{n}{k}} }

Constraint(s): s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
k = 1 , 2 , 3 , … 𝑘 1 2 3 … {\displaystyle{\displaystyle{\displaystyle k=1,2,3,\dots}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


\HurwitzZeta ⁢ @ ⁢ 1 - s ⁢ h k = 2 ⁢ Γ ⁡ ( s ) ( 2 ⁢ π ⁢ k ) s ⁢ ∑ r = 1 k cos ⁡ ( π ⁢ s 2 - 2 ⁢ π ⁢ r ⁢ h k ) ⁢ \HurwitzZeta ⁢ @ ⁢ s ⁢ r k \HurwitzZeta @ 1 𝑠 ℎ 𝑘 2 Euler-Gamma 𝑠 superscript 2 𝑘 𝑠 superscript subscript 𝑟 1 𝑘 𝑠 2 2 𝑟 ℎ 𝑘 \HurwitzZeta @ 𝑠 𝑟 𝑘 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{1-s}{\frac{h}{k}}=% \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)\HurwitzZeta@{s}{\frac{r}{k}}}}} {\displaystyle \HurwitzZeta@{1-s}{\frac{h}{k}} = \frac{2 \EulerGamma@{s}}{(2 \cpi k)^s} \* \sum_{r=1}^k \cos@{\frac{\cpi s}{2} - \frac{2 \cpi r h}{k}} \HurwitzZeta@{s}{\frac{r}{k}} }

Constraint(s): s ≠ 0 , 1 𝑠 0 1 {\displaystyle{\displaystyle{\displaystyle s\neq 0,1}}} &
h , k ℎ 𝑘 {\displaystyle{\displaystyle{\displaystyle h,k}}} integers, 1 ≤ h ≤ k 1 ℎ 𝑘 {\displaystyle{\displaystyle{\displaystyle 1\leq h\leq k}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


Derivatives

a-Derivative

∂ ∂ ⁡ a ⁡ \HurwitzZeta ⁢ @ ⁢ s ⁢ a = - s ⁢ \HurwitzZeta ⁢ @ ⁢ s + 1 ⁢ a partial-derivative 𝑎 \HurwitzZeta @ 𝑠 𝑎 𝑠 \HurwitzZeta @ 𝑠 1 𝑎 {\displaystyle{\displaystyle{\displaystyle\frac{\partial}{\partial a}% \HurwitzZeta@{s}{a}=-s\HurwitzZeta@{s+1}{a}}}} {\displaystyle \pderiv{}{a} \HurwitzZeta@{s}{a} = -s \HurwitzZeta@{s+1}{a} }

Constraint(s): s ≠ 0 , 1 𝑠 0 1 {\displaystyle{\displaystyle{\displaystyle s\neq 0,1}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


s-Derivatives

\HurwitzZeta ′ ⁢ @ ⁢ 0 ⁢ a = ln ⁡ Γ ⁡ ( a ) - 1 2 ⁢ ln ⁡ ( 2 ⁢ π ) superscript \HurwitzZeta ′ @ 0 𝑎 Euler-Gamma 𝑎 1 2 2 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{0}{a}=\ln% \Gamma\left(a\right)-\tfrac{1}{2}\ln\left(2\pi\right)}}} {\displaystyle \HurwitzZeta'@{0}{a} = \ln@@{\EulerGamma@{a}} - \tfrac{1}{2} \ln@{2\cpi} }

Constraint(s): a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a>0}}}


\HurwitzZeta ′ ⁢ @ ⁢ s ⁢ a = - ln ⁡ a a s ⁢ ( 1 2 + a s - 1 ) - a 1 - s ( s - 1 ) 2 + s ⁢ ( s + 1 ) ⁢ ∫ 0 ∞ \PeriodicBernoulliB ⁢ 2 ⁢ @ ⁢ x ⁢ ln ⁡ ( x + a ) ( x + a ) s + 2 ⁢ d x - ( 2 ⁢ s + 1 ) ⁢ ∫ 0 ∞ \PeriodicBernoulliB ⁢ 2 ⁢ @ ⁢ x ( x + a ) s + 2 ⁢ d x superscript \HurwitzZeta ′ @ 𝑠 𝑎 𝑎 superscript 𝑎 𝑠 1 2 𝑎 𝑠 1 superscript 𝑎 1 𝑠 superscript 𝑠 1 2 𝑠 𝑠 1 superscript subscript 0 \PeriodicBernoulliB 2 @ 𝑥 𝑥 𝑎 superscript 𝑥 𝑎 𝑠 2 𝑥 2 𝑠 1 superscript subscript 0 \PeriodicBernoulliB 2 @ 𝑥 superscript 𝑥 𝑎 𝑠 2 𝑥 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{s}{a}=-\frac{% \ln a}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{(s-1)^{2}}+% s(s+1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB{2}@{x}\ln\left(x+a\right)}{(x% +a)^{s+2}}\mathrm{d}x-(2s+1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB{2}@{x}}% {(x+a)^{s+2}}\mathrm{d}x}}} {\displaystyle \HurwitzZeta'@{s}{a} = -\frac{\ln@@{a}}{a^s} \left( \frac{1}{2} + \frac{a}{s-1} \right) - \frac{a^{1-s}}{(s-1)^2} + s (s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} \ln@{x+a}}{(x+a)^{s+2}} \diff{x} - (2s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x}}{(x+a)^{s+2}} \diff{x} }

Constraint(s): ℜ ⁡ s > - 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>-1}}} &
s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a>0}}}


( - 1 ) k ⁢ \HurwitzZeta ( k ) ⁢ @ ⁢ s ⁢ a = ( ln ⁡ a ) k a s ⁢ ( 1 2 + a s - 1 ) + k ! ⁢ a 1 - s ⁢ ∑ r = 0 k - 1 ( ln ⁡ a ) r r ! ⁢ ( s - 1 ) k - r + 1 - s ⁢ ( s + 1 ) ⁢ ∫ 0 ∞ \PeriodicBernoulliB ⁢ 2 ⁢ @ ⁢ x ⁢ ( ln ⁡ ( x + a ) ) k ( x + a ) s + 2 ⁢ d x + k ⁢ ( 2 ⁢ s + 1 ) ⁢ ∫ 0 ∞ \PeriodicBernoulliB ⁢ 2 ⁢ @ ⁢ x ⁢ ( ln ⁡ ( x + a ) ) k - 1 ( x + a ) s + 2 ⁢ d x - k ⁢ ( k - 1 ) ⁢ ∫ 0 ∞ \PeriodicBernoulliB ⁢ 2 ⁢ @ ⁢ x ⁢ ( ln ⁡ ( x + a ) ) k - 2 ( x + a ) s + 2 ⁢ d x superscript 1 𝑘 superscript \HurwitzZeta 𝑘 @ 𝑠 𝑎 superscript 𝑎 𝑘 superscript 𝑎 𝑠 1 2 𝑎 𝑠 1 𝑘 superscript 𝑎 1 𝑠 superscript subscript 𝑟 0 𝑘 1 superscript 𝑎 𝑟 𝑟 superscript 𝑠 1 𝑘 𝑟 1 𝑠 𝑠 1 superscript subscript 0 \PeriodicBernoulliB 2 @ 𝑥 superscript 𝑥 𝑎 𝑘 superscript 𝑥 𝑎 𝑠 2 𝑥 𝑘 2 𝑠 1 superscript subscript 0 \PeriodicBernoulliB 2 @ 𝑥 superscript 𝑥 𝑎 𝑘 1 superscript 𝑥 𝑎 𝑠 2 𝑥 𝑘 𝑘 1 superscript subscript 0 \PeriodicBernoulliB 2 @ 𝑥 superscript 𝑥 𝑎 𝑘 2 superscript 𝑥 𝑎 𝑠 2 𝑥 {\displaystyle{\displaystyle{\displaystyle(-1)^{k}\HurwitzZeta^{(k)}@{s}{a}=% \frac{(\ln a)^{k}}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum_% {r=0}^{k-1}\frac{(\ln a)^{r}}{r!(s-1)^{k-r+1}}-s(s+1)\int_{0}^{\infty}\frac{% \PeriodicBernoulliB{2}@{x}(\ln\left(x+a\right))^{k}}{(x+a)^{s+2}}\mathrm{d}x+k% (2s+1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB{2}@{x}(\ln\left(x+a\right))^{% k-1}}{(x+a)^{s+2}}\mathrm{d}x-k(k-1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB% {2}@{x}(\ln\left(x+a\right))^{k-2}}{(x+a)^{s+2}}\mathrm{d}x}}} {\displaystyle \opminus^k \HurwitzZeta^{(k)}@{s}{a} = \frac{(\ln@@{a})^k}{a^s} \left( \frac{1}{2} + \frac{a}{s-1} \right) + k! a^{1-s} \sum_{r=0}^{k-1} \frac{(\ln@@{a})^r}{r! (s-1)^{k-r+1}} - s (s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} (\ln@{x+a})^k} {(x+a)^{s+2}} \diff{x} + k (2s+1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} (\ln@{x+a})^{k-1}} {(x+a)^{s+2}} \diff{x} - k (k-1) \int_0^\infty \frac{\PeriodicBernoulliB{2}@{x} (\ln@{x+a})^{k-2}} {(x+a)^{s+2}} \diff{x} }

Constraint(s): ℜ ⁡ s > - 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>-1}}} &
s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a>0}}}


\HurwitzZeta ′ ⁢ @ ⁢ 1 - 2 ⁢ n ⁢ h k = ( ψ ⁡ ( 2 ⁢ n ) - ln ⁡ ( 2 ⁢ π ⁢ k ) ) ⁢ \BernoulliB ⁢ 2 ⁢ n ⁢ @ ⁢ h / k 2 ⁢ n - ( ψ ⁡ ( 2 ⁢ n ) - ln ⁡ ( 2 ⁢ π ) ) ⁢ \BernoulliB ⁢ 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 ) ⁢ \HurwitzZeta ′ ⁢ @ ⁢ 2 ⁢ n ⁢ r k + \RiemannZeta ′ ⁢ @ ⁢ 1 - 2 ⁢ n k 2 ⁢ n superscript \HurwitzZeta ′ @ 1 2 𝑛 ℎ 𝑘 digamma 2 𝑛 2 𝑘 \BernoulliB 2 𝑛 @ ℎ 𝑘 2 𝑛 digamma 2 𝑛 2 \BernoulliB 2 𝑛 2 𝑛 superscript 𝑘 2 𝑛 superscript 1 𝑛 1 superscript 2 𝑘 2 𝑛 superscript subscript 𝑟 1 𝑘 1 2 𝑟 ℎ 𝑘 digamma 2 𝑛 1 𝑟 𝑘 ⋅ superscript 1 𝑛 1 2 2 𝑛 1 superscript 2 𝑘 2 𝑛 superscript subscript 𝑟 1 𝑘 1 2 𝑟 ℎ 𝑘 superscript \HurwitzZeta ′ @ 2 𝑛 𝑟 𝑘 superscript \RiemannZeta ′ @ 1 2 𝑛 superscript 𝑘 2 𝑛 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{1-2n}{\frac{h% }{k}}=\frac{(\psi\left(2n\right)-\ln\left(2\pi k\right))\BernoulliB{2n}@{h/k}}% {2n}-\frac{(\psi\left(2n\right)-\ln\left(2\pi\right))\BernoulliB{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)\HurwitzZeta% ^{\prime}@{2n}{\frac{r}{k}}+\frac{\RiemannZeta^{\prime}@{1-2n}}{k^{2n}}}}} {\displaystyle \HurwitzZeta'@{1-2n}{\frac{h}{k}} = \frac{(\digamma@{2n} - \ln@{2 \cpi k}) \BernoulliB{2n}@{h/k}}{2n} - \frac{(\digamma@{2n} - \ln@{2 \cpi}) \BernoulliB{2n}}{2 n k^{2n}} + \frac{\opminus^{n+1} \cpi}{(2 \cpi k)^{2n}} \sum_{r=1}^{k-1} \sin@{\frac{2 \cpi r h}{k}} \digamma^{(2n-1)}@{\frac{r}{k}} + \frac{\opminus^{n+1} 2 \cdot (2n-1)!}{(2 \cpi k)^{2n}} \sum_{r=1}^{k-1} \cos@{\frac{2 \cpi r h}{k}} \HurwitzZeta'@{2n}{\frac{r}{k}} + \frac{\RiemannZeta'@{1-2n}}{k^{2n}} }

Constraint(s): h , k ℎ 𝑘 {\displaystyle{\displaystyle{\displaystyle h,k}}} are integers with 1 ≤ h ≤ k 1 ℎ 𝑘 {\displaystyle{\displaystyle{\displaystyle 1\leq h\leq k}}} &
n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}


\HurwitzZeta ′ ⁢ @ ⁢ 1 - 2 ⁢ n ⁢ 1 2 = - \BernoulliB ⁢ 2 ⁢ n ⁢ ln ⁡ 2 n ⋅ 4 n - ( 2 2 ⁢ n - 1 - 1 ) ⁢ \RiemannZeta ′ ⁢ @ ⁢ 1 - 2 ⁢ n 2 2 ⁢ n - 1 superscript \HurwitzZeta ′ @ 1 2 𝑛 1 2 \BernoulliB 2 𝑛 2 ⋅ 𝑛 superscript 4 𝑛 superscript 2 2 𝑛 1 1 superscript \RiemannZeta ′ @ 1 2 𝑛 superscript 2 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{1-2n}{\tfrac{% 1}{2}}=-\frac{\BernoulliB{2n}\ln 2}{n\cdot 4^{n}}-\frac{(2^{2n-1}-1)% \RiemannZeta^{\prime}@{1-2n}}{2^{2n-1}}}}} {\displaystyle \HurwitzZeta'@{1-2n}{\tfrac{1}{2}} = - \frac{\BernoulliB{2n} \ln@@{2}}{n \cdot 4^n} - \frac{(2^{2n-1}-1) \RiemannZeta'@{1-2n}}{2^{2n-1}} }

Constraint(s): n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}


\HurwitzZeta ′ ⁢ @ ⁢ 1 - 2 ⁢ n ⁢ 1 3 = - π ⁢ ( 9 n - 1 ) ⁢ \BernoulliB ⁢ 2 ⁢ n 8 ⁢ n ⁢ 3 ⁢ ( 3 2 ⁢ n - 1 - 1 ) - \BernoulliB ⁢ 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 ) ⁢ \RiemannZeta ′ ⁢ @ ⁢ 1 - 2 ⁢ n 2 ⋅ 3 2 ⁢ n - 1 superscript \HurwitzZeta ′ @ 1 2 𝑛 1 3 superscript 9 𝑛 1 \BernoulliB 2 𝑛 8 𝑛 3 superscript 3 2 𝑛 1 1 \BernoulliB 2 𝑛 3 ⋅ 4 𝑛 superscript 3 2 𝑛 1 superscript 1 𝑛 digamma 2 𝑛 1 1 3 2 3 superscript 6 2 𝑛 1 superscript 3 2 𝑛 1 1 superscript \RiemannZeta ′ @ 1 2 𝑛 ⋅ 2 superscript 3 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{1-2n}{\tfrac{% 1}{3}}=-\frac{\pi(9^{n}-1)\BernoulliB{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{% \BernoulliB{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)% \RiemannZeta^{\prime}@{1-2n}}{2\cdot 3^{2n-1}}}}} {\displaystyle \HurwitzZeta'@{1-2n}{\tfrac{1}{3}} = -\frac{\cpi (9^n-1) \BernoulliB{2n}}{8n \sqrt{3} (3^{2n-1}-1)} - \frac{\BernoulliB{2n} \ln@@{3}}{4n \cdot 3^{2n-1}} - \frac{\opminus^n \digamma^{(2n-1)}@{\frac{1}{3}}}{2 \sqrt{3} (6\cpi)^{2n-1}} - \frac{\left( 3^{2n-1}-1 \right) \RiemannZeta'@{1-2n}}{2 \cdot 3^{2n-1}} }

Constraint(s): n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}


∑ r = 1 k - 1 \HurwitzZeta ′ ⁢ @ ⁢ s ⁢ r k = ( k s - 1 ) ⁢ \RiemannZeta ′ ⁢ @ ⁢ s + k s ⁢ \RiemannZeta ⁢ @ ⁢ s ⁢ ln ⁡ k superscript subscript 𝑟 1 𝑘 1 superscript \HurwitzZeta ′ @ 𝑠 𝑟 𝑘 superscript 𝑘 𝑠 1 superscript \RiemannZeta ′ @ 𝑠 superscript 𝑘 𝑠 \RiemannZeta @ 𝑠 𝑘 {\displaystyle{\displaystyle{\displaystyle\sum_{r=1}^{k-1}\HurwitzZeta^{\prime% }@{s}{\frac{r}{k}}=(k^{s}-1)\RiemannZeta^{\prime}@{s}+k^{s}\RiemannZeta@{s}\ln k% }}} {\displaystyle \sum_{r \hiderel{=} 1}^{k-1} \HurwitzZeta'@{s}{\frac{r}{k}} = (k^s - 1) \RiemannZeta'@{s} + k^s \RiemannZeta@{s} \ln@@{k} }

Constraint(s): s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
k = 1 , 2 , 3 , … 𝑘 1 2 3 … {\displaystyle{\displaystyle{\displaystyle k=1,2,3,\dots}}}


Integral Representations

\HurwitzZeta ⁢ @ ⁢ s ⁢ a = 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 ⁢ e - a ⁢ x 1 - e - x ⁢ d x \HurwitzZeta @ 𝑠 𝑎 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 𝑎 𝑥 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{1}{\Gamma% \left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}{\mathrm{e}^{-ax}}}{1-{\mathrm{e}% ^{-x}}}\mathrm{d}x}}} {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-ax}}{1-\expe^{-x}} \diff{x} }

Constraint(s): ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


\HurwitzZeta ⁢ @ ⁢ s ⁢ a = - s ⁢ ∫ - a ∞ x - ⌊ x ⌋ - 1 2 ( x + a ) s + 1 ⁢ d x \HurwitzZeta @ 𝑠 𝑎 𝑠 superscript subscript 𝑎 𝑥 𝑥 1 2 superscript 𝑥 𝑎 𝑠 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=-s\int_{-a}^{% \infty}\frac{x-\left\lfloor x\right\rfloor-\frac{1}{2}}{(x+a)^{s+1}}\mathrm{d}% x}}} {\displaystyle \HurwitzZeta@{s}{a} = -s \int_{-a}^\infty \frac{x-\floor{x}-\frac{1}{2}}{(x+a)^{s+1}} \diff{x} }

Constraint(s): - 1 < ℜ ⁡ s < 0 1 𝑠 0 {\displaystyle{\displaystyle{\displaystyle-1<\Re{s}<0}}} &
0 < a ≤ 1 0 𝑎 1 {\displaystyle{\displaystyle{\displaystyle 0<a\leq 1}}}


\HurwitzZeta ⁢ @ ⁢ 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 \HurwitzZeta @ 𝑠 𝑎 1 2 superscript 𝑎 𝑠 superscript 𝑎 1 𝑠 𝑠 1 1 Euler-Gamma 𝑠 superscript subscript 0 1 𝑥 1 1 𝑥 1 2 superscript 𝑥 𝑠 1 𝑎 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{1}{2}a^{-s% }+\frac{a^{1-s}}{s-1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(% \frac{1}{{\mathrm{e}^{x}}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{{% \mathrm{e}^{ax}}}\mathrm{d}x}}} {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{2} a^{-s} + \frac{a^{1-s}}{s-1} + \frac{1}{\EulerGamma@{s}} \int_0^\infty \left( \frac{1}{\expe^x-1} - \frac{1}{x}+ \frac{1}{2} \right) \frac{x^{s-1}}{\expe^{ax}} \diff{x} }

Constraint(s): ℜ ⁡ s > - 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>-1}}} &
s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


\HurwitzZeta ⁢ @ ⁢ s ⁢ a = 1 2 ⁢ a - s + a 1 - s s - 1 + ∑ k = 1 n Γ ⁡ ( s + 2 ⁢ k - 1 ) Γ ⁡ ( s ) ⁢ \BernoulliB ⁢ 2 ⁢ k ( 2 ⁢ k ) ! ⁢ a - 2 ⁢ k - s + 1 + 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ ( 1 e x - 1 - 1 x + 1 2 - ∑ k = 1 n \BernoulliB ⁢ 2 ⁢ k ( 2 ⁢ k ) ! ⁢ x 2 ⁢ k - 1 ) ⁢ x s - 1 ⁢ e - a ⁢ x ⁢ d x \HurwitzZeta @ 𝑠 𝑎 1 2 superscript 𝑎 𝑠 superscript 𝑎 1 𝑠 𝑠 1 superscript subscript 𝑘 1 𝑛 Euler-Gamma 𝑠 2 𝑘 1 Euler-Gamma 𝑠 \BernoulliB 2 𝑘 2 𝑘 superscript 𝑎 2 𝑘 𝑠 1 1 Euler-Gamma 𝑠 superscript subscript 0 1 𝑥 1 1 𝑥 1 2 superscript subscript 𝑘 1 𝑛 \BernoulliB 2 𝑘 2 𝑘 superscript 𝑥 2 𝑘 1 superscript 𝑥 𝑠 1 𝑎 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{1}{2}a^{-s% }+\frac{a^{1-s}}{s-1}+\sum_{k=1}^{n}\frac{\Gamma\left(s+2k-1\right)}{\Gamma% \left(s\right)}\frac{\BernoulliB{2k}}{(2k)!}a^{-2k-s+1}+\frac{1}{\Gamma\left(s% \right)}\int_{0}^{\infty}\left(\frac{1}{{\mathrm{e}^{x}}-1}-\frac{1}{x}+\frac{% 1}{2}-\sum_{k=1}^{n}\frac{\BernoulliB{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}{% \mathrm{e}^{-ax}}\mathrm{d}x}}} {\displaystyle \HurwitzZeta@{s}{a} = \frac{1}{2} a^{-s}+ \frac{a^{1-s}}{s-1} + \sum_{k=1}^n \frac{\EulerGamma@{s+2k-1}}{\EulerGamma@{s}} \frac{\BernoulliB{2k}}{(2k)!} a^{-2k-s+1} + \frac{1}{\EulerGamma@{s}} \int_0^\infty \left( \frac{1}{\expe^x-1} - \frac{1}{x} + \frac{1}{2} - \sum_{k=1}^n \frac{\BernoulliB{2k}}{(2k)!} x^{2k-1} \right) x^{s-1} \expe^{-ax} \diff{x} }

Constraint(s): ℜ ⁡ s > - ( 2 ⁢ n + 1 ) 𝑠 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>-(2n+1)}}} &
s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


\HurwitzZeta ⁢ @ ⁢ 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 \HurwitzZeta @ 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 2 𝑥 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\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}({\mathrm{e}^{2\pi x}}-1)}\mathrm{% d}x}}} {\displaystyle \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} (\expe^{2 \cpi x}-1)} \diff{x} }

Constraint(s): s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


\HurwitzZeta ⁢ @ ⁢ s ⁢ a = Γ ⁡ ( 1 - s ) 2 ⁢ π ⁢ i ⁢ ∫ - ∞ ( 0 + ) e a ⁢ z ⁢ z s - 1 1 - e z ⁢ d z \HurwitzZeta @ 𝑠 𝑎 Euler-Gamma 1 𝑠 2 imaginary-unit superscript subscript limit-from 0 𝑎 𝑧 superscript 𝑧 𝑠 1 1 𝑧 𝑧 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}=\frac{\Gamma% \left(1-s\right)}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}\frac{{\mathrm{e}^{az}}z% ^{s-1}}{1-{\mathrm{e}^{z}}}\mathrm{d}z}}} {\displaystyle \HurwitzZeta@{s}{a} = \frac{\EulerGamma@{1-s}}{2 \cpi \iunit} \int_{-\infty}^{(0+)} \frac{\expe^{az} z^{s-1}}{1 - \expe^z} \diff{z} }

Constraint(s): s ≠ 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}} &

the integration contour is a loop around the negative real axis; it starts at - ∞ {\displaystyle{\displaystyle{\displaystyle-\infty}}} , encircles the origin once in the positive direction without enclosing any of the points

z = ± 2 ⁢ π ⁢ i 𝑧 plus-or-minus 2 imaginary-unit {\displaystyle{\displaystyle{\displaystyle z=\pm 2\pi\mathrm{i}}}} , ± 4 ⁢ π ⁢ i , … , plus-or-minus 4 imaginary-unit … {\displaystyle{\displaystyle{\displaystyle\pm 4\pi\mathrm{i},\ldots,}}} and returns to - ∞ {\displaystyle{\displaystyle{\displaystyle-\infty}}}


Further Integral Representations

1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 ⁢ e - a ⁢ x 2 ⁢ cosh ⁡ x ⁢ d x = 4 - s ⁢ ( \HurwitzZeta ⁢ @ ⁢ s ⁢ 1 4 + 1 4 ⁢ a - \HurwitzZeta ⁢ @ ⁢ s ⁢ 3 4 + 1 4 ⁢ a ) 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 𝑎 𝑥 2 𝑥 𝑥 superscript 4 𝑠 \HurwitzZeta @ 𝑠 1 4 1 4 𝑎 \HurwitzZeta @ 𝑠 3 4 1 4 𝑎 {\displaystyle{\displaystyle{\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{% 0}^{\infty}\frac{x^{s-1}{\mathrm{e}^{-ax}}}{2\cosh x}\mathrm{d}x=4^{-s}\left(% \HurwitzZeta@{s}{\tfrac{1}{4}+\tfrac{1}{4}a}-\HurwitzZeta@{s}{\tfrac{3}{4}+% \tfrac{1}{4}a}\right)}}} {\displaystyle \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-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) }

Constraint(s): ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}} &
ℜ ⁡ a > - 1 𝑎 1 {\displaystyle{\displaystyle{\displaystyle\Re{a}>-1}}}


∫ 0 a x n ⁢ ψ ⁡ ( x ) ⁢ d x = ( - 1 ) n - 1 ⁢ \RiemannZeta ′ ⁢ @ - n + ( - 1 ) n ⁢ H n ⁢ \BernoulliB ⁢ n + 1 n + 1 - ∑ k = 0 n ( - 1 ) k ⁢ ( n k ) ⁢ h ⁢ ( k ) ⁢ \BernoulliB ⁢ k + 1 ⁢ ( a ) k + 1 ⁢ a n - k + ∑ k = 0 n ( - 1 ) k ⁢ ( n k ) ⁢ \HurwitzZeta ′ ⁢ @ - k ⁢ a ⁢ a n - k superscript subscript 0 𝑎 superscript 𝑥 𝑛 digamma 𝑥 𝑥 superscript 1 𝑛 1 superscript \RiemannZeta ′ @ 𝑛 superscript 1 𝑛 Harmonic-number 𝑛 \BernoulliB 𝑛 1 𝑛 1 superscript subscript 𝑘 0 𝑛 superscript 1 𝑘 binomial 𝑛 𝑘 ℎ 𝑘 \BernoulliB 𝑘 1 𝑎 𝑘 1 superscript 𝑎 𝑛 𝑘 superscript subscript 𝑘 0 𝑛 superscript 1 𝑘 binomial 𝑛 𝑘 superscript \HurwitzZeta ′ @ 𝑘 𝑎 superscript 𝑎 𝑛 𝑘 {\displaystyle{\displaystyle{\displaystyle\int_{0}^{a}x^{n}\psi\left(x\right)% \mathrm{d}x=(-1)^{n-1}\RiemannZeta^{\prime}@{-n}+(-1)^{n}H_{n}\frac{% \BernoulliB{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}h(k)% \frac{\BernoulliB{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.% 0pt}{}{n}{k}\HurwitzZeta^{\prime}@{-k}{a}a^{n-k}}}} {\displaystyle \int_0^a x^n \digamma@{x} \diff{x} = \opminus^{n-1} \RiemannZeta'@{-n} + \opminus^n \HarmonicNumber{n} \frac{\BernoulliB{n+1}}{n+1} - \sum_{k=0}^n \opminus^k \binom{n}{k} h(k) \frac{\BernoulliB{k+1}(a)}{k+1} a^{n-k} + \sum_{k=0}^n \opminus^k \binom{n}{k} \HurwitzZeta'@{-k}{a} a^{n-k} }

Substitution(s): H n = ∑ k = 1 n k - 1 Harmonic-number 𝑛 superscript subscript 𝑘 1 𝑛 superscript 𝑘 1 {\displaystyle{\displaystyle{\displaystyle{\displaystyle H_{n}=\sum_{k=1}^{n}k% ^{-1}}}}}


Constraint(s): n = 1 , 2 , … 𝑛 1 2 … {\displaystyle{\displaystyle{\displaystyle n=1,2,\dots}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


n ⁢ ∫ 0 a \HurwitzZeta ′ ⁢ @ ⁢ 1 - n ⁢ x ⁢ d x = \HurwitzZeta ′ ⁢ @ - n ⁢ a - \RiemannZeta ′ ⁢ @ - n + \BernoulliB ⁢ n + 1 - \BernoulliB ⁢ n + 1 ⁢ @ ⁢ a n ⁢ ( n + 1 ) 𝑛 superscript subscript 0 𝑎 superscript \HurwitzZeta ′ @ 1 𝑛 𝑥 𝑥 superscript \HurwitzZeta ′ @ 𝑛 𝑎 superscript \RiemannZeta ′ @ 𝑛 \BernoulliB 𝑛 1 \BernoulliB 𝑛 1 @ 𝑎 𝑛 𝑛 1 {\displaystyle{\displaystyle{\displaystyle n\int_{0}^{a}\HurwitzZeta^{\prime}@% {1-n}{x}\mathrm{d}x=\HurwitzZeta^{\prime}@{-n}{a}-\RiemannZeta^{\prime}@{-n}+% \frac{\BernoulliB{n+1}-\BernoulliB{n+1}@{a}}{n(n+1)}}}} {\displaystyle n \int_0^a \HurwitzZeta'@{1-n}{x} \diff{x} = \HurwitzZeta'@{-n}{a} - \RiemannZeta'@{-n} + \frac{\BernoulliB{n+1} - \BernoulliB{n+1}@{a}}{n(n+1)} }

Constraint(s): n = 1 , 2 , … 𝑛 1 2 … {\displaystyle{\displaystyle{\displaystyle n=1,2,\dots}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}}


Further Series Representations

∑ n = 0 ∞ ( - 1 ) n ( n + a ) s = 1 Γ ⁡ ( s ) ⁢ ∫ 0 ∞ x s - 1 ⁢ e - a ⁢ x 1 + e - x ⁢ d x = 2 - s ⁢ ( \HurwitzZeta ⁢ @ ⁢ s ⁢ 1 2 ⁢ a - \HurwitzZeta ⁢ @ ⁢ s ⁢ 1 2 ⁢ ( 1 + a ) ) superscript subscript 𝑛 0 superscript 1 𝑛 superscript 𝑛 𝑎 𝑠 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 𝑎 𝑥 1 𝑥 𝑥 superscript 2 𝑠 \HurwitzZeta @ 𝑠 1 2 𝑎 \HurwitzZeta @ 𝑠 1 2 1 𝑎 {\displaystyle{\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}{\mathrm% {e}^{-ax}}}{1+{\mathrm{e}^{-x}}}\mathrm{d}x=2^{-s}\left(\HurwitzZeta@{s}{% \tfrac{1}{2}a}-\HurwitzZeta@{s}{\tfrac{1}{2}(1+a)}\right)}}} {\displaystyle \sum_{n \hiderel{=} 0}^\infty \frac{\opminus^n}{(n+a)^s} \hiderel{=} \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1} \expe^{-ax}}{1+\expe^{-x}} \diff{x} \hiderel{=} 2^{-s} \left( \HurwitzZeta@{s}{\tfrac{1}{2}a} - \HurwitzZeta@{s}{\tfrac{1}{2} (1+a)} \right) }

Constraint(s): ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}} , ℜ ⁡ s > 0 𝑠 0 {\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}} &
or ℜ ⁡ a = 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}=0}}} , ℑ ⁡ a ≠ 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Im{a}\neq 0}}} , 0 < ℜ ⁡ s < 1 0 𝑠 1 {\displaystyle{\displaystyle{\displaystyle 0<\Re{s}<1}}}


∑ n = 1 ∞ χ ⁢ ( n ) n s = k - s ⁢ ∑ r = 1 k χ ⁢ ( r ) ⁢ \HurwitzZeta ⁢ @ ⁢ s ⁢ r k superscript subscript 𝑛 1 𝜒 𝑛 superscript 𝑛 𝑠 superscript 𝑘 𝑠 superscript subscript 𝑟 1 𝑘 𝜒 𝑟 \HurwitzZeta @ 𝑠 𝑟 𝑘 {\displaystyle{\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\chi(n)}{n^% {s}}=k^{-s}\sum_{r=1}^{k}\chi(r)\HurwitzZeta@{s}{\frac{r}{k}}}}} {\displaystyle \sum_{n \hiderel{=} 1}^\infty \frac{\chi(n)}{n^s} = k^{-s} \sum_{r=1}^k \chi(r) \HurwitzZeta@{s}{\frac{r}{k}} }

Constraint(s): ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}} &
χ ⁢ ( n ) 𝜒 𝑛 {\displaystyle{\displaystyle{\displaystyle\chi(n)}}} is a Dirichlet character~ ( mod k ) pmod 𝑘 {\displaystyle{\displaystyle{\displaystyle\pmod{k}}}}


Sums

∑ k = 1 ∞ ( - 1 ) k k ⁢ \HurwitzZeta ⁢ @ ⁢ 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 \HurwitzZeta @ 𝑛 𝑘 𝑎 𝑛 Euler-Gamma 𝑎 superscript subscript product 𝑗 0 𝑛 1 Euler-Gamma 𝑎 2 𝑗 1 imaginary-unit 𝑛 {\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k% }\HurwitzZeta@{nk}{a}=-n\ln\Gamma\left(a\right)+\ln\left(\prod_{j=0}^{n-1}% \Gamma\left(a-{\mathrm{e}^{(2j+1)\pi\mathrm{i}/n}}\right)\right)}}} {\displaystyle \sum_{k \hiderel{=} 1}^\infty \frac{\opminus^k}{k} \HurwitzZeta@{nk}{a} = -n \ln@@{\EulerGamma@{a}} + \ln@{\prod_{j=0}^{n-1} \EulerGamma@{a-\expe^{(2j+1)\cpi \iunit/n}}} }

Constraint(s): n = 2 , 3 , 4 , … 𝑛 2 3 4 … {\displaystyle{\displaystyle{\displaystyle n=2,3,4,\dots}}} &
ℜ ⁡ a ≥ 1 𝑎 1 {\displaystyle{\displaystyle{\displaystyle\Re{a}\geq 1}}}


∑ k = 1 ∞ ( n + k k ) ⁢ \HurwitzZeta ⁢ @ ⁢ n + k + 1 ⁢ a ⁢ z k = ( - 1 ) n n ! ⁢ ( ψ ( n ) ⁡ ( a ) - ψ ( n ) ⁡ ( a - z ) ) superscript subscript 𝑘 1 binomial 𝑛 𝑘 𝑘 \HurwitzZeta @ 𝑛 𝑘 1 𝑎 superscript 𝑧 𝑘 superscript 1 𝑛 𝑛 digamma 𝑛 𝑎 digamma 𝑛 𝑎 𝑧 {\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\genfrac{(}{)}{0.% 0pt}{}{n+k}{k}\HurwitzZeta@{n+k+1}{a}z^{k}=\frac{(-1)^{n}}{n!}\left({\psi^{(n)% }}\left(a\right)-{\psi^{(n)}}\left(a-z\right)\right)}}} {\displaystyle \sum_{k \hiderel{=} 1}^\infty \binom{n+k}{k} \HurwitzZeta@{n+k+1}{a} z^k = \frac{\opminus^n}{n!} \left( \digamma^{(n)}@{a} - \digamma^{(n)}@{a-z} \right) }

Constraint(s): n = 1 , 2 , 3 , … 𝑛 1 2 3 … {\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}} &
ℜ ⁡ a > 0 𝑎 0 {\displaystyle{\displaystyle{\displaystyle\Re{a}>0}}} &
| z | < | a | 𝑧 𝑎 {\displaystyle{\displaystyle{\displaystyle|z|<|a|}}}


∑ k = 2 ∞ k 2 k ⁢ \HurwitzZeta ⁢ @ ⁢ k + 1 ⁤ 3 4 = 8 ⁢ C superscript subscript 𝑘 2 𝑘 superscript 2 𝑘 \HurwitzZeta @ 𝑘 1 3 4 8 𝐶 {\displaystyle{\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{k}{2^{k}}% \HurwitzZeta@{k+1}{\tfrac{3}{4}}=8C}}} {\displaystyle \sum_{k \hiderel{=} 2}^\infty \frac{k}{2^k} \HurwitzZeta@{k+1}{\tfrac{3}{4}} = 8\CatalansConstant }

a-Asymptotic Behavior

\HurwitzZeta ⁢ @ ⁢ s ⁢ a + 1 = \RiemannZeta ⁢ @ ⁢ s - s ⁢ \RiemannZeta ⁢ @ ⁢ s + 1 ⁢ a + \BigO ⁢ @ ⁢ a 2 \HurwitzZeta @ 𝑠 𝑎 1 \RiemannZeta @ 𝑠 𝑠 \RiemannZeta @ 𝑠 1 𝑎 \BigO @ superscript 𝑎 2 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a+1}=\RiemannZeta@{% s}-s\RiemannZeta@{s+1}a+\BigO@{a^{2}}}}} {\displaystyle \HurwitzZeta@{s}{a+1} = \RiemannZeta@{s} - s \RiemannZeta@{s+1} a + \BigO@{a^2} }

Constraint(s): a → 0 → 𝑎 0 {\displaystyle{\displaystyle{\displaystyle a\to 0}}} with s 𝑠 {\displaystyle{\displaystyle{\displaystyle s}}} ( ≠ 1 ) absent 1 {\displaystyle{\displaystyle{\displaystyle(\neq 1)}}} fixed


\HurwitzZeta ⁢ @ ⁢ s ⁢ α + i ⁢ β → 0 → \HurwitzZeta @ 𝑠 𝛼 imaginary-unit 𝛽 0 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{\alpha+\mathrm{i}% \beta}\to 0}}} {\displaystyle \HurwitzZeta@{s}{\alpha+\iunit\beta} \to 0 }

Constraint(s): β → ± ∞ → 𝛽 plus-or-minus {\displaystyle{\displaystyle{\displaystyle\beta\to\pm\infty}}} with s 𝑠 {\displaystyle{\displaystyle{\displaystyle s}}} fixed &
ℜ ⁡ s > 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re s>1}}}


\HurwitzZeta ⁢ @ ⁢ s ⁢ a - a 1 - s s - 1 - 1 2 ⁢ a - s ∼ ∑ k = 1 ∞ \BernoulliB ⁢ 2 ⁢ k ( 2 ⁢ k ) ! ⁢ Γ ⁡ ( s + 2 ⁢ k - 1 ) Γ ⁡ ( s ) ⁢ a 1 - s - 2 ⁢ k similar-to \HurwitzZeta @ 𝑠 𝑎 superscript 𝑎 1 𝑠 𝑠 1 1 2 superscript 𝑎 𝑠 superscript subscript 𝑘 1 \BernoulliB 2 𝑘 2 𝑘 Euler-Gamma 𝑠 2 𝑘 1 Euler-Gamma 𝑠 superscript 𝑎 1 𝑠 2 𝑘 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta@{s}{a}-\frac{a^{1-s}}{s% -1}-\frac{1}{2}a^{-s}\sim\sum_{k=1}^{\infty}\frac{\BernoulliB{2k}}{(2k)!}\frac% {\Gamma\left(s+2k-1\right)}{\Gamma\left(s\right)}a^{1-s-2k}}}} {\displaystyle \HurwitzZeta@{s}{a} - \frac{a^{1-s}}{s-1} - \frac{1}{2} a^{-s} \sim \sum_{k=1}^{\infty} \frac{\BernoulliB{2k}}{(2k)!} \frac{\EulerGamma@{s+2k-1}}{\EulerGamma@{s}} a^{1-s-2k} }

Constraint(s): a → ∞ → 𝑎 {\displaystyle{\displaystyle{\displaystyle a\to\infty}}} in the sector | \ph ⁢ @ ⁢ @ ⁢ a | ≤ π - δ ( < π ) \ph @ @ 𝑎 annotated 𝛿 absent {\displaystyle{\displaystyle{\displaystyle|\ph@@{a}|\leq\pi-\delta(<\pi)}}} , with s ( ≠ 1 ) annotated 𝑠 absent 1 {\displaystyle{\displaystyle{\displaystyle s(\neq 1)}}} and δ 𝛿 {\displaystyle{\displaystyle{\displaystyle\delta}}} fixed


\HurwitzZeta ′ ⁢ @ - 1 ⁢ a - 1 12 + 1 4 ⁢ a 2 - ( 1 12 - 1 2 ⁢ a + 1 2 ⁢ a 2 ) ⁢ ln ⁡ a ∼ - ∑ k = 1 ∞ \BernoulliB ⁢ 2 ⁢ k + 2 ( 2 ⁢ k + 2 ) ⁢ ( 2 ⁢ k + 1 ) ⁢ 2 ⁢ k ⁢ a - 2 ⁢ k similar-to superscript \HurwitzZeta ′ @ 1 𝑎 1 12 1 4 superscript 𝑎 2 1 12 1 2 𝑎 1 2 superscript 𝑎 2 𝑎 superscript subscript 𝑘 1 \BernoulliB 2 𝑘 2 2 𝑘 2 2 𝑘 1 2 𝑘 superscript 𝑎 2 𝑘 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{-1}{a}-\frac{% 1}{12}+\frac{1}{4}a^{2}-\left(\frac{1}{12}-\frac{1}{2}a+\frac{1}{2}a^{2}\right% )\ln a\sim-\sum_{k=1}^{\infty}\frac{\BernoulliB{2k+2}}{(2k+2)(2k+1)2k}a^{-2k}}}} {\displaystyle \HurwitzZeta'@{-1}{a} - \frac{1}{12} + \frac{1}{4} a^2 - \left( \frac{1}{12} - \frac{1}{2} a + \frac{1}{2} a^2 \right) \ln@@{a} \sim -\sum_{k=1}^\infty \frac{\BernoulliB{2k+2}}{(2k+2)(2k+1)2k} a^{-2k} }

Constraint(s): a → ∞ → 𝑎 {\displaystyle{\displaystyle{\displaystyle a\to\infty}}} in the sector | \ph ⁢ @ ⁢ @ ⁢ a | ≤ 1 2 ⁢ π - δ ( < 1 2 ⁢ π ) \ph @ @ 𝑎 annotated 1 2 𝛿 absent 1 2 {\displaystyle{\displaystyle{\displaystyle|\ph@@{a}|\leq\tfrac{1}{2}\pi-\delta% (<\tfrac{1}{2}\pi)}}}


\HurwitzZeta ′ ⁢ @ - 2 ⁢ a - 1 12 ⁢ a + 1 9 ⁢ a 3 - ( 1 6 ⁢ a - 1 2 ⁢ a 2 + 1 3 ⁢ a 3 ) ⁢ ln ⁡ a ∼ ∑ k = 1 ∞ 2 ⁢ \BernoulliB ⁢ 2 ⁢ k + 2 ( 2 ⁢ k + 2 ) ⁢ ( 2 ⁢ k + 1 ) ⁢ 2 ⁢ k ⁢ ( 2 ⁢ k - 1 ) ⁢ a - ( 2 ⁢ k - 1 ) similar-to superscript \HurwitzZeta ′ @ 2 𝑎 1 12 𝑎 1 9 superscript 𝑎 3 1 6 𝑎 1 2 superscript 𝑎 2 1 3 superscript 𝑎 3 𝑎 superscript subscript 𝑘 1 2 \BernoulliB 2 𝑘 2 2 𝑘 2 2 𝑘 1 2 𝑘 2 𝑘 1 superscript 𝑎 2 𝑘 1 {\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{-2}{a}-\frac{% 1}{12}a+\frac{1}{9}a^{3}-\left(\frac{1}{6}a-\frac{1}{2}a^{2}+\frac{1}{3}a^{3}% \right)\ln a\sim\sum_{k=1}^{\infty}\frac{2\!\BernoulliB{2k+2}}{(2k+2)(2k+1)2k(% 2k-1)}a^{-(2k-1)}}}} {\displaystyle \HurwitzZeta'@{-2}{a} - \frac{1}{12} a + \frac{1}{9} a^3 - \left( \frac{1}{6} a - \frac{1}{2} a^2 + \frac{1}{3} a^3 \right) \ln@@{a} \sim \sum_{k=1}^\infty \frac{2\!\BernoulliB{2k+2}}{(2k+2)(2k+1)2k(2k-1)} a^{-(2k-1)} }

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