# §25.11 Hurwitz Zeta Function

## §25.11(i) Definition

The function $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ was introduced in Hurwitz (1882) and defined by the series expansion

 25.11.1 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\sum_{n=0}^{\infty}\frac{1}{(n+a)^% {s}},$ $\Re{s}>1$, $a\neq 0,-1,-2,\dots$. Defines: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function Symbols: $\Re{}$: real part, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Referenced by: §25.11(i), §25.11(v), §25.11(vi) Permalink: http://dlmf.nist.gov/25.11.E1 Encodings: TeX, pMML, png See also: Annotations for 25.11(i)

$\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ has a meromorphic continuation in the $s$-plane, its only singularity in $\mathbb{C}$ being a simple pole at $s=1$ with residue $1$. As a function of $a$, with $s$ ($\neq 1$) fixed, $\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ is analytic in the half-plane $\Re{a}>0$. The Riemann zeta function is a special case:

 25.11.2 $\mathop{\zeta\/}\nolimits\!\left(s,1\right)=\mathop{\zeta\/}\nolimits\!\left(s% \right).$

For most purposes it suffices to restrict $0<\Re{a}\leq 1$ because of the following straightforward consequences of (25.11.1):

 25.11.3 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\mathop{\zeta\/}\nolimits\!\left(s% ,a+1\right)+a^{-s},$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $a$: real or complex parameter and $s$: complex variable Permalink: http://dlmf.nist.gov/25.11.E3 Encodings: TeX, pMML, png See also: Annotations for 25.11(i)
 25.11.4 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\mathop{\zeta\/}\nolimits\!\left(s% ,a+m\right)+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{s}},$ $m=1,2,3,\dots$.

Most references treat real $a$ with $0.

## §25.11(iii) Representations by the Euler–Maclaurin Formula

 25.11.5 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\sum_{n=0}^{N}\frac{1}{(n+a)^{s}}+% \frac{(N+a)^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}% {(x+a)^{s+1}}\mathrm{d}x,$ $s\neq 1$, $\Re{s}>0$, $a>0$, $N=0,1,2,3,\dots$.
 25.11.6 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{1}{a^{s}}\left(\frac{1}{2}+% \frac{a}{s-1}\right)-\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{\mathop{\widetilde% {B}_{2}\/}\nolimits\!\left(x\right)-B_{2}}{(x+a)^{s+2}}\mathrm{d}x,$ $s\neq 1$, $\Re{s}>-1$, $a>0$. Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\mathop{\zeta\/}\nolimits\!\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\widetilde{B}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: periodic Bernoulli functions, $\Re{}$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Referenced by: Equations (25.11.6), (25.11.19), and (25.11.20) Permalink: http://dlmf.nist.gov/25.11.E6 Encodings: TeX, pMML, png Errata (effective with 1.0.12): Originally the integrand was incorrect because its numerator contained the function $\mathop{\widetilde{B}_{2}\/}\nolimits\!\left(x\right)$. The correct function is $\frac{\mathop{\widetilde{B}_{2}\/}\nolimits\!\left(x\right)-B_{2}}{2}$. Reported 2016-05-08 by Clemens Heuberger See also: Annotations for 25.11(iii)
 25.11.7 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{1}{a^{s}}+\frac{1}{(1+a)^{s}% }\left(\frac{1}{2}+\frac{1+a}{s-1}\right)+\sum_{k=1}^{n}\binom{s+2k-2}{2k-1}% \frac{B_{2k}}{2k}\frac{1}{(1+a)^{s+2k-1}}-\binom{s+2n}{2n+1}\int_{1}^{\infty}% \frac{\mathop{\widetilde{B}_{2n+1}\/}\nolimits\!\left(x\right)}{(x+a)^{s+2n+1}% }\mathrm{d}x,$ $s\neq 1$, $a>0$, $n=1,2,3,\dots$, $\Re{s}>-2n$.

For $\mathop{\widetilde{B}_{n}\/}\nolimits\!\left(x\right)$ see §24.2(iii).

## §25.11(iv) Series Representations

 25.11.8 $\mathop{\zeta\/}\nolimits\!\left(s,\tfrac{1}{2}a\right)=\mathop{\zeta\/}% \nolimits\!\left(s,\tfrac{1}{2}a+\tfrac{1}{2}\right)+2^{s}\sum_{n=0}^{\infty}% \frac{(-1)^{n}}{(n+a)^{s}},$ $\Re{s}>0$, $s\neq 1$, $0.
 25.11.9 $\mathop{\zeta\/}\nolimits\!\left(1-s,a\right)=\frac{2\mathop{\Gamma\/}% \nolimits\!\left(s\right)}{(2\pi)^{s}}\*\sum_{n=1}^{\infty}\frac{1}{n^{s}}% \mathop{\cos\/}\nolimits\!\left(\tfrac{1}{2}\pi s-2n\pi a\right),$ $\Re{s}>1$, $0.
 25.11.10 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\sum_{n=0}^{\infty}\frac{{\left(s% \right)_{n}}}{n!}\mathop{\zeta\/}\nolimits\!\left(n+s\right)(1-a)^{n},$ $s\neq 1$, $|a-1|<1$. Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{\zeta\/}\nolimits\!\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Referenced by: §25.11(iv), §25.11(iv), §25.11(iv), Other Changes Permalink: http://dlmf.nist.gov/25.11.E10 Encodings: TeX, pMML, png Notational Change (effective with 1.0.9): We have rewritten the original summation $\sum_{n=0}^{\infty}\frac{\mathop{\Gamma\/}\nolimits\!\left(n+s\right)}{n!% \mathop{\Gamma\/}\nolimits\!\left(s\right)}\mathop{\zeta\/}\nolimits\!\left(n+% s\right)(1-a)^{n}$ more concisely as $\sum_{n=0}^{\infty}\frac{{\left(s\right)_{n}}}{n!}\mathop{\zeta\/}\nolimits\!% \left(n+s\right)(1-a)^{n}$ using the Pochhammer symbol. Reported 2014-05-22 See also: Annotations for 25.11(iv)

When $a=\frac{1}{2}$, (25.11.10) reduces to (25.8.3); compare (25.11.11).

For other series expansions similar to (25.11.10) see Coffey (2008).

## §25.11(v) Special Values

Throughout this subsection $\Re{a}>0$.

 25.11.11 $\mathop{\zeta\/}\nolimits\!\left(s,\tfrac{1}{2}\right)=(2^{s}-1)\mathop{\zeta% \/}\nolimits\!\left(s\right),$ $s\neq 1$. Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\mathop{\zeta\/}\nolimits\!\left(\NVar{s}\right)$: Riemann zeta function and $s$: complex variable Referenced by: §25.11(iv), §25.11(v) Permalink: http://dlmf.nist.gov/25.11.E11 Encodings: TeX, pMML, png See also: Annotations for 25.11(v)
 25.11.12 $\mathop{\zeta\/}\nolimits\!\left(n+1,a\right)=\frac{(-1)^{n+1}{\mathop{\psi\/}% \nolimits^{(n)}}\!\left(a\right)}{n!},$ $n=1,2,3,\dots$.
 25.11.13 $\mathop{\zeta\/}\nolimits\!\left(0,a\right)=\tfrac{1}{2}-a.$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function and $a$: real or complex parameter Permalink: http://dlmf.nist.gov/25.11.E13 Encodings: TeX, pMML, png See also: Annotations for 25.11(v)
 25.11.14 $\mathop{\zeta\/}\nolimits\!\left(-n,a\right)=-\frac{\mathop{B_{n+1}\/}% \nolimits\!\left(a\right)}{n+1},$ $n=0,1,2,\dots$.
 25.11.15 $\mathop{\zeta\/}\nolimits\!\left(s,ka\right)=k^{-s}\*\sum_{n=0}^{k-1}\mathop{% \zeta\/}\nolimits\!\left(s,a+\frac{n}{k}\right),$ $s\neq 1$, $k=1,2,3,\dots$.
 25.11.16 $\mathop{\zeta\/}\nolimits\!\left(1-s,\frac{h}{k}\right)=\frac{2\mathop{\Gamma% \/}\nolimits\!\left(s\right)}{(2\pi k)^{s}}\*\sum_{r=1}^{k}\mathop{\cos\/}% \nolimits\!\left(\frac{\pi s}{2}-\frac{2\pi rh}{k}\right)\mathop{\zeta\/}% \nolimits\!\left(s,\frac{r}{k}\right),$ $s\neq 0,1$; $h,k$ integers, $1\leq h\leq k$.

## §25.11(vi) Derivatives

### $a$-Derivative

 25.11.17 $\frac{\partial}{\partial a}\mathop{\zeta\/}\nolimits\!\left(s,a\right)=-s% \mathop{\zeta\/}\nolimits\!\left(s+1,a\right),$ $s\neq 0,1$; $\Re{a}>0$.

### $s$-Derivatives

In (25.11.18)–(25.11.24) primes on $\mathop{\zeta\/}\nolimits$ denote derivatives with respect to $s$. Similarly in §§25.11(viii) and 25.11(xii).

 25.11.18 $\mathop{\zeta\/}\nolimits'\!\left(0,a\right)=\mathop{\ln\/}\nolimits\mathop{% \Gamma\/}\nolimits\!\left(a\right)-\tfrac{1}{2}\mathop{\ln\/}\nolimits\!\left(% 2\pi\right),$ $a>0$.
 25.11.19 $\mathop{\zeta\/}\nolimits'\!\left(s,a\right)=-\frac{\mathop{\ln\/}\nolimits a}% {a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-\frac{a^{1-s}}{(s-1)^{2}}+\frac{% s(s+1)}{2}\int_{0}^{\infty}\frac{(\mathop{\widetilde{B}_{2}\/}\nolimits\!\left% (x\right)-B_{2})\mathop{\ln\/}\nolimits\!\left(x+a\right)}{(x+a)^{s+2}}\mathrm% {d}x-\frac{(2s+1)}{2}\int_{0}^{\infty}\frac{\mathop{\widetilde{B}_{2}\/}% \nolimits\!\left(x\right)-B_{2}}{(x+a)^{s+2}}\mathrm{d}x,$ $\Re{s}>-1$, $s\neq 1$, $a>0$. Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\mathop{\zeta\/}\nolimits\!\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\mathop{\widetilde{B}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: periodic Bernoulli functions, $\Re{}$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Referenced by: Equations (25.11.6), (25.11.19), and (25.11.20) Permalink: http://dlmf.nist.gov/25.11.E19 Encodings: TeX, pMML, png Errata (effective with 1.0.12): Originally both integrands were incorrect because their numerators contained the function $\mathop{\widetilde{B}_{2}\/}\nolimits\!\left(x\right)$. The correct function is $\frac{\mathop{\widetilde{B}_{2}\/}\nolimits\!\left(x\right)-B_{2}}{2}$. Reported 2016-06-27 by Gergő Nemes See also: Annotations for 25.11(vi)
 25.11.20 $(-1)^{k}{\mathop{\zeta\/}\nolimits^{(k)}}\!\left(s,a\right)=\frac{(\mathop{\ln% \/}\nolimits a)^{k}}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)+k!a^{1-s}% \sum_{r=0}^{k-1}\frac{(\mathop{\ln\/}\nolimits a)^{r}}{r!(s-1)^{k-r+1}}-\frac{% s(s+1)}{2}\int_{0}^{\infty}\frac{(\mathop{\widetilde{B}_{2}\/}\nolimits\!\left% (x\right)-B_{2})(\mathop{\ln\/}\nolimits\!\left(x+a\right))^{k}}{(x+a)^{s+2}}% \mathrm{d}x+\frac{k(2s+1)}{2}\int_{0}^{\infty}\frac{(\mathop{\widetilde{B}_{2}% \/}\nolimits\!\left(x\right)-B_{2})(\mathop{\ln\/}\nolimits\!\left(x+a\right))% ^{k-1}}{(x+a)^{s+2}}\mathrm{d}x-\frac{k(k-1)}{2}\int_{0}^{\infty}\frac{(% \mathop{\widetilde{B}_{2}\/}\nolimits\!\left(x\right)-B_{2})(\mathop{\ln\/}% \nolimits\!\left(x+a\right))^{k-2}}{(x+a)^{s+2}}\mathrm{d}x,$ $\Re{s}>-1$, $s\neq 1$, $a>0$. Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\mathop{\zeta\/}\nolimits\!\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\mathrm{d}\NVar{x}$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\mathop{\widetilde{B}_{\NVar{n}}\/}\nolimits\!\left(\NVar{x}\right)$: periodic Bernoulli functions, $\Re{}$: real part, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $k$: integer Referenced by: Equations (25.11.6), (25.11.19), and (25.11.20) Permalink: http://dlmf.nist.gov/25.11.E20 Encodings: TeX, pMML, png Errata (effective with 1.0.12): Originally all three integrands were incorrect because their numerators contained the function $\mathop{\widetilde{B}_{2}\/}\nolimits\!\left(x\right)$. The correct function is $\frac{\mathop{\widetilde{B}_{2}\/}\nolimits\!\left(x\right)-B_{2}}{2}$. Reported 2016-06-27 by Gergő Nemes See also: Annotations for 25.11(vi)
 25.11.21 $\mathop{\zeta\/}\nolimits'\!\left(1-2n,\frac{h}{k}\right)=\frac{(\mathop{\psi% \/}\nolimits\!\left(2n\right)-\mathop{\ln\/}\nolimits\!\left(2\pi k\right))% \mathop{B_{2n}\/}\nolimits\!\left(h/k\right)}{2n}-\frac{(\mathop{\psi\/}% \nolimits\!\left(2n\right)-\mathop{\ln\/}\nolimits\!\left(2\pi\right))B_{2n}}{% 2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\mathop{\sin\/}% \nolimits\!\left(\frac{2\pi rh}{k}\right){\mathop{\psi\/}\nolimits^{(2n-1)}}\!% \left(\frac{r}{k}\right)+\frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=% 1}^{k-1}\mathop{\cos\/}\nolimits\!\left(\frac{2\pi rh}{k}\right)\mathop{\zeta% \/}\nolimits'\!\left(2n,\frac{r}{k}\right)+\frac{\mathop{\zeta\/}\nolimits'\!% \left(1-2n\right)}{k^{2n}},$

where $h,k$ are integers with $1\leq h\leq k$ and $n=1,2,3,\dots$.

 25.11.22 $\mathop{\zeta\/}\nolimits'\!\left(1-2n,\tfrac{1}{2}\right)=-\frac{B_{2n}% \mathop{\ln\/}\nolimits 2}{n\cdot 4^{n}}-\frac{(2^{2n-1}-1)\mathop{\zeta\/}% \nolimits'\!\left(1-2n\right)}{2^{2n-1}},$ $n=1,2,3,\dots$.
 25.11.23 $\mathop{\zeta\/}\nolimits'\!\left(1-2n,\tfrac{1}{3}\right)=-\frac{\pi(9^{n}-1)% B_{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{B_{2n}\mathop{\ln\/}\nolimits 3}{4n\cdot 3% ^{2n-1}}-\frac{(-1)^{n}{\mathop{\psi\/}\nolimits^{(2n-1)}}\!\left(\frac{1}{3}% \right)}{2\sqrt{3}(6\pi)^{2n-1}}-\frac{\left(3^{2n-1}-1\right)\mathop{\zeta\/}% \nolimits'\!\left(1-2n\right)}{2\cdot 3^{2n-1}},$ $n=1,2,3,\dots$.
 25.11.24 $\sum_{r=1}^{k-1}\mathop{\zeta\/}\nolimits'\!\left(s,\frac{r}{k}\right)=(k^{s}-% 1)\mathop{\zeta\/}\nolimits'\!\left(s\right)+k^{s}\mathop{\zeta\/}\nolimits\!% \left(s\right)\mathop{\ln\/}\nolimits k,$ $s\neq 1$, $k=1,2,3,\dots$.

## §25.11(vii) Integral Representations

 25.11.25 $\displaystyle\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ $\displaystyle=\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int_{0}^{% \infty}\frac{x^{s-1}e^{-ax}}{1-e^{-x}}\mathrm{d}x,$ $\Re{s}>1$, $\Re{a}>0$. 25.11.26 $\displaystyle\mathop{\zeta\/}\nolimits\!\left(s,a\right)$ $\displaystyle=-s\int_{-a}^{\infty}\frac{x-\left\lfloor x\right\rfloor-\frac{1}% {2}}{(x+a)^{s+1}}\mathrm{d}x,$ $-1<\Re{s}<0$, $0.
 25.11.27 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s% -1}+\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int_{0}^{\infty}\left% (\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{ax}}\mathrm% {d}x,$ $\Re{s}>-1$, $s\neq 1$, $\Re{a}>0$.
 25.11.28 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s% -1}+\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}+\frac{% 1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{% e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}x^{2k-1}% \right)x^{s-1}e^{-ax}\mathrm{d}x,$ $\Re{s}>-(2n+1)$, $s\neq 1$, $\Re{a}>0$. Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{\zeta\/}\nolimits\!\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $!$: factorial (as in $n!$), $\int$: integral, $\Re{}$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Referenced by: §25.11(vii), Other Changes Permalink: http://dlmf.nist.gov/25.11.E28 Encodings: TeX, pMML, png Notational Change (effective with 1.0.9): We have rewritten the original summation $\sum_{k=1}^{n}\frac{\mathop{\Gamma\/}\nolimits\!\left(s+2k-1\right)}{\mathop{% \Gamma\/}\nolimits\!\left(s\right)}\frac{B_{2k}}{(2k)!}a^{-2k-s+1}$ more concisely as $\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}$ using the Pochhammer symbol. Reported 2014-05-22 See also: Annotations for 25.11(vii)
 25.11.29 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s% -1}+2\int_{0}^{\infty}\frac{\mathop{\sin\/}\nolimits\!\left(s\mathop{\mathrm{% arctan}\/}\nolimits\!\left(x/a\right)\right)}{(a^{2}+x^{2})^{s/2}(e^{2\pi x}-1% )}\mathrm{d}x,$ $s\neq 1$, $\Re{a}>0$.
 25.11.30 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)=\frac{\mathop{\Gamma\/}\nolimits\!% \left(1-s\right)}{2\pi i}\int_{-\infty}^{(0+)}\frac{e^{az}z^{s-1}}{1-e^{z}}% \mathrm{d}z,$ $s\neq 1$, $\Re{a}>0$,

where the integration contour is a loop around the negative real axis as described for (25.5.20).

## §25.11(viii) Further Integral Representations

 25.11.31 $\frac{1}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}\int_{0}^{\infty}\frac{x^{% s-1}e^{-ax}}{2\mathop{\cosh\/}\nolimits x}\mathrm{d}x=4^{-s}\left(\mathop{% \zeta\/}\nolimits\!\left(s,\tfrac{1}{4}+\tfrac{1}{4}a\right)-\mathop{\zeta\/}% \nolimits\!\left(s,\tfrac{3}{4}+\tfrac{1}{4}a\right)\right),$ $\Re{s}>0$, $\Re{a}>-1$.
 25.11.32 $\int_{0}^{a}x^{n}\mathop{\psi\/}\nolimits\!\left(x\right)\mathrm{d}x=(-1)^{n-1% }\mathop{\zeta\/}\nolimits'\!\left(-n\right)+(-1)^{n}h(n)\frac{B_{n+1}}{n+1}-% \sum_{k=0}^{n}(-1)^{k}\binom{n}{k}h(k)\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}% ^{n}(-1)^{k}\binom{n}{k}\mathop{\zeta\/}\nolimits'\!\left(-k,a\right)a^{n-k},$ $n=1,2,\dots$, $\Re{a}>0$,

where

 25.11.33 $h(n)=\sum_{k=1}^{n}k^{-1}.$ Symbols: $k$: nonnegative integer, $n$: nonnegative integer and $h(n)$: sum Referenced by: §25.16(ii) Permalink: http://dlmf.nist.gov/25.11.E33 Encodings: TeX, pMML, png See also: Annotations for 25.11(viii)
 25.11.34 $n\int_{0}^{a}\mathop{\zeta\/}\nolimits'\!\left(1-n,x\right)\mathrm{d}x=\mathop% {\zeta\/}\nolimits'\!\left(-n,a\right)-\mathop{\zeta\/}\nolimits'\!\left(-n% \right)+\frac{B_{n+1}-\mathop{B_{n+1}\/}\nolimits\!\left(a\right)}{n(n+1)},$ $n=1,2,\dots$, $\Re{a}>0$.

## §25.11(ix) Integrals

See Prudnikov et al. (1990, §2.3), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).

## §25.11(x) Further Series Representations

 25.11.35 $\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=\frac{1}{\mathop{\Gamma\/}% \nolimits\!\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}% \mathrm{d}x=2^{-s}\left(\mathop{\zeta\/}\nolimits\!\left(s,\tfrac{1}{2}a\right% )-\mathop{\zeta\/}\nolimits\!\left(s,\tfrac{1}{2}(1+a)\right)\right),$ $\Re{a}>0$, $\Re{s}>0$; or $\Re{a}=0$, $\Im{a}\neq 0$, $0<\Re{s}<1$.

When $a=1$, (25.11.35) reduces to (25.2.3).

 25.11.36 $\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}=k^{-s}\sum_{r=1}^{k}\chi(r)\mathop{% \zeta\/}\nolimits\!\left(s,\frac{r}{k}\right),$ $\Re{s}>1$,

where $\chi(n)$ is a Dirichlet character $\pmod{k}$27.8).

## §25.11(xi) Sums

 25.11.37 $\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\mathop{\zeta\/}\nolimits\!\left(nk,a% \right)=-n\mathop{\ln\/}\nolimits\mathop{\Gamma\/}\nolimits\!\left(a\right)+% \mathop{\ln\/}\nolimits\!\left(\prod_{j=0}^{n-1}\mathop{\Gamma\/}\nolimits\!% \left(a-e^{(2j+1)\pi i/n}\right)\right),$ $n=2,3,4,\dots$, $\Re{a}\geq 1$.
 25.11.38 $\sum_{k=1}^{\infty}\binom{n+k}{k}\mathop{\zeta\/}\nolimits\!\left(n+k+1,a% \right)z^{k}=\frac{(-1)^{n}}{n!}\left({\mathop{\psi\/}\nolimits^{(n)}}\!\left(% a\right)-{\mathop{\psi\/}\nolimits^{(n)}}\!\left(a-z\right)\right),$ $n=1,2,3,\dots$, $\Re{a}>0$, $|z|<|a|$.
 25.11.39 $\sum_{k=2}^{\infty}\frac{k}{2^{k}}\mathop{\zeta\/}\nolimits\!\left(k+1,\tfrac{% 3}{4}\right)=8G,$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $k$: nonnegative integer and $G$: Catalan’s constant Permalink: http://dlmf.nist.gov/25.11.E39 Encodings: TeX, pMML, png See also: Annotations for 25.11(xi)

where $G$ is Catalan’s constant:

 25.11.40 $G=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}=0.91596\;55941\;772\dots.$ Symbols: $n$: nonnegative integer and $G$: Catalan’s constant Notes: For more digits see OEIS Sequence A006752; see also Sloane (2003). Permalink: http://dlmf.nist.gov/25.11.E40 Encodings: TeX, pMML, png See also: Annotations for 25.11(xi)

For further sums see Prudnikov et al. (1990, pp. 396–397) and Hansen (1975, pp. 358–360).

## §25.11(xii) $a$-Asymptotic Behavior

As $a\to 0$ with $s$ $(\neq 1)$ fixed,

 25.11.41 $\mathop{\zeta\/}\nolimits\!\left(s,a+1\right)=\mathop{\zeta\/}\nolimits\!\left% (s\right)-s\mathop{\zeta\/}\nolimits\!\left(s+1\right)a+\mathop{O\/}\nolimits% \!\left(a^{2}\right).$

As $\beta\to\pm\infty$ with $s$ fixed, $\Re{s}>1$,

 25.11.42 $\mathop{\zeta\/}\nolimits\!\left(s,\alpha+i\beta\right)\to 0,$ Symbols: $\mathop{\zeta\/}\nolimits\!\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function and $s$: complex variable Referenced by: §25.11(xii) Permalink: http://dlmf.nist.gov/25.11.E42 Encodings: TeX, pMML, png See also: Annotations for 25.11(xii)

uniformly with respect to bounded nonnegative values of $\alpha$.

As $a\to\infty$ in the sector $|\mathop{\mathrm{ph}\/}\nolimits a|\leq\pi-\delta(<\pi)$, with $s(\neq 1)$ and $\delta$ fixed, we have the asymptotic expansion

 25.11.43 $\mathop{\zeta\/}\nolimits\!\left(s,a\right)-\frac{a^{1-s}}{s-1}-\frac{1}{2}a^{% -s}\sim\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}.$ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\mathop{\Gamma\/}\nolimits\!\left(\NVar{z}\right)$: gamma function, $\mathop{\zeta\/}\nolimits\!\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $k$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Referenced by: §25.11(xii), §25.11(xii), Other Changes Permalink: http://dlmf.nist.gov/25.11.E43 Encodings: TeX, pMML, png Notational Change (effective with 1.0.9): We have rewritten the original summation $\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}\frac{\mathop{\Gamma\/}\nolimits\!\left% (s+2k-1\right)}{\mathop{\Gamma\/}\nolimits\!\left(s\right)}a^{1-s-2k}$ more concisely as $\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}$ using the Pochhammer symbol. Reported 2014-05-22 See also: Annotations for 25.11(xii)

Similarly, as $a\to\infty$ in the sector $|\mathop{\mathrm{ph}\/}\nolimits a|\leq\frac{1}{2}\pi-\delta(<\frac{1}{2}\pi)$,

 25.11.44 $\mathop{\zeta\/}\nolimits'\!\left(-1,a\right)-\frac{1}{12}+\frac{1}{4}a^{2}-% \left(\frac{1}{12}-\frac{1}{2}a+\frac{1}{2}a^{2}\right)\mathop{\ln\/}\nolimits a% \sim-\sum_{k=1}^{\infty}\frac{B_{2k+2}}{(2k+2)(2k+1)2k}a^{-2k},$

and

 25.11.45 $\mathop{\zeta\/}\nolimits'\!\left(-2,a\right)-\frac{1}{12}a+\frac{1}{9}a^{3}-% \left(\frac{1}{6}a-\frac{1}{2}a^{2}+\frac{1}{3}a^{3}\right)\mathop{\ln\/}% \nolimits a\sim\sum_{k=1}^{\infty}\frac{2\!B_{2k+2}}{(2k+2)(2k+1)2k(2k-1)}a^{-% (2k-1)}.$

For the more general case $\mathop{\zeta\/}\nolimits'\!\left(-m,a\right)$, $m=1,2,\dots$, see Elizalde (1986).

For an exponentially-improved form of (25.11.43) see Paris (2005b).