Hurwitz Zeta Function

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Hurwitz Zeta Function

Polylogarithms >>

Definition

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,1\right)=\zeta\left(s\right)}}}$
${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\zeta\left(s,a+1\right)+a^{-s}}}}$
${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\zeta\left(s,a+m\right)+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{s}}}}}$

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

Representations by the Euler-Maclaurin Formula

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{N}\frac{1}{(n+a)^{s}}+\frac{(N+a)^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\floor{x}}{(x+a)^{s+1}}\diffd x}}}$

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

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-s(s+1)\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x\right)}{(x+a)^{s+2}}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{s}>-1}}}$ &
${\displaystyle{\displaystyle{\displaystyle a>0}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{a^{s}}+\frac{1}{(1+a)^{s}}\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum_{k=1}^{n}\binom{s+2k-2}{2k-1}\frac{B_{2k}}{2k}\frac{1}{(1+a)^{s+2k-1}}-\binom{s+2n}{2n+1}\int_{1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{(x+a)^{s+2n+1}}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle a>0}}}$ &
${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{s}>-2n}}}$

Series Representations

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,\tfrac{1}{2}a\right)=\zeta\left(s,\tfrac{1}{2}a+\tfrac{1}{2}\right)+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}}$ &
${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle 0<a\leq 1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(1-s,a\right)=\frac{2\Gamma\left(s\right)}{(2\cpi)^{s}}\*\sum_{n=1}^{\infty}\frac{1}{n^{s}}\cos\!\left(\tfrac{1}{2}\cpi s-2n\cpi a\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{\Gamma\left(n+s\right)}{n!\Gamma\left(s\right)}\zeta\left(n+s\right)(1-a)^{n}}}}$

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

Special Values

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,\tfrac{1}{2}\right)=(2^{s}-1)\zeta\left(s\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle\zeta\left(n+1,a\right)=\frac{(-1)^{n+1}{\psi^{(n)}}\left(a\right)}{n!}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(0,a\right)=\tfrac{1}{2}-a}}}$

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

${\displaystyle{\displaystyle{\displaystyle\zeta\left(-n,a\right)=-\frac{B_{n+1}\left(a\right)}{n+1}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=0,1,2,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,ka\right)=k^{-s}\*\sum_{n=0}^{k-1}\zeta\left(s,a+\frac{n}{k}\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle k=1,2,3,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(1-s,\frac{h}{k}\right)=\frac{2\Gamma\left(s\right)}{(2\cpi k)^{s}}\*\sum_{r=1}^{k}\cos\!\left(\frac{\cpi s}{2}-\frac{2\cpi rh}{k}\right)\zeta\left(s,\frac{r}{k}\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 0,1}}}$ &
${\displaystyle{\displaystyle{\displaystyle h,k}}}$ integers, ${\displaystyle{\displaystyle{\displaystyle 1\leq h\leq k}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$

Derivatives

a-Derivative

${\displaystyle{\displaystyle{\displaystyle\frac{\partial}{\partial a}\zeta\left(s,a\right)=-s\zeta\left(s+1,a\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 0,1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$

s-Derivatives

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{0}{a}=\ln\Gamma\left(a\right)-\tfrac{1}{2}\ln\left(2\pi\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{s}{a}=-\frac{\ln a}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{(s-1)^{2}}+s(s+1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB{2}@{x}\ln\left(x+a\right)}{(x+a)^{s+2}}\mathrm{d}x-(2s+1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB{2}@{x}}{(x+a)^{s+2}}\mathrm{d}x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\realpart{s}>-1}}}$ &
${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle a>0}}}$

${\displaystyle{\displaystyle{\displaystyle(-1)^{k}\HurwitzZeta^{(k)}@{s}{a}=\frac{(\ln a)^{k}}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum_{r=0}^{k-1}\frac{(\ln a)^{r}}{r!(s-1)^{k-r+1}}-s(s+1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB{2}@{x}(\ln\left(x+a\right))^{k}}{(x+a)^{s+2}}\mathrm{d}x+k(2s+1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB{2}@{x}(\ln\left(x+a\right))^{k-1}}{(x+a)^{s+2}}\mathrm{d}x-k(k-1)\int_{0}^{\infty}\frac{\PeriodicBernoulliB{2}@{x}(\ln\left(x+a\right))^{k-2}}{(x+a)^{s+2}}\mathrm{d}x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\realpart{s}>-1}}}$ &
${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle a>0}}}$

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{1-2n}{\frac{h}{k}}=\frac{(\psi\left(2n\right)-\ln\left(2\pi k\right))\BernoulliB{2n}@{h/k}}{2n}-\frac{(\psi\left(2n\right)-\ln\left(2\pi\right))\BernoulliB{2n}}{2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\sin\left(\frac{2\pi rh}{k}\right){\psi^{(2n-1)}}\left(\frac{r}{k}\right)+\frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\cos\left(\frac{2\pi rh}{k}\right)\HurwitzZeta^{\prime}@{2n}{\frac{r}{k}}+\frac{\RiemannZeta^{\prime}@{1-2n}}{k^{2n}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{1-2n}{\tfrac{1}{2}}=-\frac{\BernoulliB{2n}\ln 2}{n\cdot 4^{n}}-\frac{(2^{2n-1}-1)\RiemannZeta^{\prime}@{1-2n}}{2^{2n-1}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{1-2n}{\tfrac{1}{3}}=-\frac{\pi(9^{n}-1)\BernoulliB{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{\BernoulliB{2n}\ln 3}{4n\cdot 3^{2n-1}}-\frac{(-1)^{n}{\psi^{(2n-1)}}\left(\frac{1}{3}\right)}{2\sqrt{3}(6\pi)^{2n-1}}-\frac{\left(3^{2n-1}-1\right)\RiemannZeta^{\prime}@{1-2n}}{2\cdot 3^{2n-1}}}}}$

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

${\displaystyle{\displaystyle{\displaystyle\sum_{r=1}^{k-1}\HurwitzZeta^{\prime}@{s}{\frac{r}{k}}=(k^{s}-1)\RiemannZeta^{\prime}@{s}+k^{s}\RiemannZeta@{s}\ln k}}}$

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

Integral Representations

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}\expe^{-ax}}{1-\expe^{-x}}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=-s\int_{-a}^{\infty}\frac{x-\floor{x}-\frac{1}{2}}{(x+a)^{s+1}}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle-1<\realpart{s}<0}}}$ &
${\displaystyle{\displaystyle{\displaystyle 0<a\leq 1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{\expe^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{\expe^{ax}}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\realpart{s}>-1}}}$ &
${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\sum_{k=1}^{n}\frac{\Gamma\left(s+2k-1\right)}{\Gamma\left(s\right)}\frac{B_{2k}}{(2k)!}a^{-2k-s+1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{\expe^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}\expe^{-ax}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\realpart{s}>-(2n+1)}}}$ &
${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+2\int_{0}^{\infty}\frac{\sin\!\left(s\mathrm{arctan}\!\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(\expe^{2\cpi x}-1)}\diffd x}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)=\frac{\Gamma\left(1-s\right)}{2\cpi\iunit}\int_{-\infty}^{(0+)}\frac{\expe^{az}z^{s-1}}{1-\expe^{z}}\diffd z}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$ &

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

${\displaystyle{\displaystyle{\displaystyle z=\pm 2\pi i}}}$, ${\displaystyle{\displaystyle{\displaystyle\pm 4\pi i,\ldots,}}}$ and returns to ${\displaystyle{\displaystyle{\displaystyle-\infty}}}$

Further Integral Representations

${\displaystyle{\displaystyle{\displaystyle\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}\expe^{-ax}}{2\cosh x}\diffd x=4^{-s}\left(\zeta\left(s,\tfrac{1}{4}+\tfrac{1}{4}a\right)-\zeta\left(s,\tfrac{3}{4}+\tfrac{1}{4}a\right)\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle\int_{0}^{a}x^{n}\psi\left(x\right)\mathrm{d}x=(-1)^{n-1}\RiemannZeta^{\prime}@{-n}+(-1)^{n}H_{n}\frac{\BernoulliB{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}h(k)\frac{\BernoulliB{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}{n}{k}\HurwitzZeta^{\prime}@{-k}{a}a^{n-k}}}}$

Substitution(s): ${\displaystyle{\displaystyle{\displaystyle{\displaystyle\HarmonicNumber{n}=\sum_{k=1}^{n}k^{-1}}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=1,2,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$

${\displaystyle{\displaystyle{\displaystyle n\int_{0}^{a}\HurwitzZeta^{\prime}@{1-n}{x}\mathrm{d}x=\HurwitzZeta^{\prime}@{-n}{a}-\RiemannZeta^{\prime}@{-n}+\frac{\BernoulliB{n+1}-\BernoulliB{n+1}@{a}}{n(n+1)}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=1,2,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$

Further Series Representations

${\displaystyle{\displaystyle{\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}\expe^{-ax}}{1+\expe^{-x}}\diffd x=2^{-s}\left(\zeta\left(s,\tfrac{1}{2}a\right)-\zeta\left(s,\tfrac{1}{2}(1+a)\right)\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$, ${\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}}$ &
or ${\displaystyle{\displaystyle{\displaystyle\realpart{a}=0}}}$, ${\displaystyle{\displaystyle{\displaystyle\imagpart{a}\neq 0}}}$, ${\displaystyle{\displaystyle{\displaystyle 0<\realpart{s}<1}}}$

${\displaystyle{\displaystyle{\displaystyle\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}=k^{-s}\sum_{r=1}^{k}\chi(r)\zeta\left(s,\frac{r}{k}\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\chi(n)}}}$ is a Dirichlet character~${\displaystyle{\displaystyle{\displaystyle\pmod{k}}}}$

Sums

${\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\zeta\left(nk,a\right)=-n\ln\Gamma\left(a\right)+\ln\!\left(\prod_{j=0}^{n-1}\Gamma\left(a-\expe^{(2j+1)\cpi\iunit/n}\right)\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=2,3,4,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}\geq 1}}}$

${\displaystyle{\displaystyle{\displaystyle\sum_{k=1}^{\infty}\binom{n+k}{k}\zeta\left(n+k+1,a\right)z^{k}=\frac{(-1)^{n}}{n!}\left({\psi^{(n)}}\left(a\right)-{\psi^{(n)}}\left(a-z\right)\right)}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$ &
${\displaystyle{\displaystyle{\displaystyle|z|<|a|}}}$

${\displaystyle{\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{k}{2^{k}}\zeta\left(k+1,\tfrac{3}{4}\right)=8\CatalansConstant}}}$

a-Asymptotic Behavior

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a+1\right)=\zeta\left(s\right)-s\zeta\left(s+1\right)a+O\left(a^{2}\right)}}}$

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

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,\alpha+\iunit\beta\right)\to 0}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle\beta\to\pm\infty}}}$ with ${\displaystyle{\displaystyle{\displaystyle s}}}$ fixed &
${\displaystyle{\displaystyle{\displaystyle\realpart s>1}}}$

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s,a\right)-\frac{a^{1-s}}{s-1}-\frac{1}{2}a^{-s}\sim\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}\frac{\Gamma\left(s+2k-1\right)}{\Gamma\left(s\right)}a^{1-s-2k}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a\to\infty}}}$ in the sector ${\displaystyle{\displaystyle{\displaystyle|\mathrm{ph}a|\leq\cpi-\delta(<\cpi)}}}$, with ${\displaystyle{\displaystyle{\displaystyle s(\neq 1)}}}$ and ${\displaystyle{\displaystyle{\displaystyle\delta}}}$ fixed

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{-1}{a}-\frac{1}{12}+\frac{1}{4}a^{2}-\left(\frac{1}{12}-\frac{1}{2}a+\frac{1}{2}a^{2}\right)\ln a\sim-\sum_{k=1}^{\infty}\frac{\BernoulliB{2k+2}}{(2k+2)(2k+1)2k}a^{-2k}}}}$

Constraint(s): ${\displaystyle{\displaystyle{\displaystyle a\to\infty}}}$ in the sector ${\displaystyle{\displaystyle{\displaystyle|\mathrm{ph}a|\leq\tfrac{1}{2}\cpi-\delta(<\tfrac{1}{2}\cpi)}}}$

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{-2}{a}-\frac{1}{12}a+\frac{1}{9}a^{3}-\left(\frac{1}{6}a-\frac{1}{2}a^{2}+\frac{1}{3}a^{3}\right)\ln a\sim\sum_{k=1}^{\infty}\frac{2\!\BernoulliB{2k+2}}{(2k+2)(2k+1)2k(2k-1)}a^{-(2k-1)}}}}$

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Hurwitz Zeta Function

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