# §25.11 Hurwitz Zeta Function

## §25.11(i) Definition

The function $\zeta\left(s,a\right)$ was introduced in Hurwitz (1882) and defined by the series expansion

 25.11.1 $\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}},$ $\Re s>1$, $a\neq 0,-1,-2,\dots$. ⓘ Defines: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function Symbols: $\Re$: real part, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Keywords: definition Source: Apostol (1976, p. 251) Referenced by: (25.11.11), (25.11.15), (25.11.17), (25.11.3), (25.11.4), §25.11(i), §25.11(v), §25.11(vi), (25.14.2) Permalink: http://dlmf.nist.gov/25.11.E1 Encodings: TeX, pMML, png See also: Annotations for §25.11(i), §25.11 and Ch.25

$\zeta\left(s,a\right)$ 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, $\zeta\left(s,a\right)$ is analytic in the half-plane $\Re a>0$. The Riemann zeta function is a special case:

 25.11.2 $\zeta\left(s,1\right)=\zeta\left(s\right).$ ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function and $s$: complex variable Keywords: specialization Source: Apostol (1976, p. 255) Referenced by: (25.11.10), (25.11.24), (25.12.12) Permalink: http://dlmf.nist.gov/25.11.E2 Encodings: TeX, pMML, png See also: Annotations for §25.11(i), §25.11 and Ch.25

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 $\zeta\left(s,a\right)=\zeta\left(s,a+1\right)+a^{-s},$ ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Keywords: recurrence Source: Derivable from (25.11.1) by comparing its $n=0$ term with its sum over $n\geq 1$. Permalink: http://dlmf.nist.gov/25.11.E3 Encodings: TeX, pMML, png See also: Annotations for §25.11(i), §25.11 and Ch.25
 25.11.4 $\zeta\left(s,a\right)=\zeta\left(s,a+m\right)+\sum_{n=0}^{m-1}\frac{1}{(n+a)^{% s}},$ $m=1,2,3,\dots$. ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $m$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Keywords: recurrence Source: Derivable from (25.11.1) by comparing its sum over $n=0,\ldots,m-1$, $m\geq 1$, with its sum over $n\geq m$. Permalink: http://dlmf.nist.gov/25.11.E4 Encodings: TeX, pMML, png See also: Annotations for §25.11(i), §25.11 and Ch.25

Most references treat real $a$ with $0.

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

 25.11.5 $\zeta\left(s,a\right)=\sum_{n=0}^{N}\frac{1}{(n+a)^{s}}+\frac{(N+a)^{1-s}}{s-1% }-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{(x+a)^{s+1}}\mathrm{d% }x,$ $s\neq 1$, $\Re s>0$, $a>0$, $N=0,1,2,3,\dots$.
 25.11.6 $\zeta\left(s,a\right)=\frac{1}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-% \frac{s(s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{% (x+a)^{s+2}}\mathrm{d}x,$ $s\neq 1$, $\Re s>-1$, $a>0$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$: periodic Bernoulli functions, $\Re$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: Euler–Maclaurin formula, improper integral Source: Apostol (1985a, (25), p. 231) Referenced by: Erratum 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 $\widetilde{B}_{2}\left(x\right)$. The correct function is $\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{2}$. Reported 2016-05-08 by Clemens Heuberger See also: Annotations for §25.11(iii), §25.11 and Ch.25
 25.11.7 $\zeta\left(s,a\right)=\frac{1}{a^{s}}+\frac{1}{(1+a)^{s}}\left(\frac{1}{2}+% \frac{1+a}{s-1}\right)+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}% \frac{B_{2k}}{2k}\frac{1}{(1+a)^{s+2k-1}}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}% \int_{1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{(x+a)^{s+2n+1}}% \mathrm{d}x,$ $s\neq 1$, $a>0$, $n=1,2,3,\dots$, $\Re s>-2n$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$: binomial coefficient, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$: periodic Bernoulli functions, $\Re$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: Euler–Maclaurin formula, improper integral Source: Derivable from (25.11.5) by setting $N=1$ and repeatedly integrating by parts using (1.2.6), (24.2.2), (24.2.4), (24.2.11), (24.2.12), (24.4.34). Referenced by: §25.11(iii) Permalink: http://dlmf.nist.gov/25.11.E7 Encodings: TeX, pMML, png See also: Annotations for §25.11(iii), §25.11 and Ch.25

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

## §25.11(iv) Series Representations

 25.11.8 $\zeta\left(s,\tfrac{1}{2}a\right)=\zeta\left(s,\tfrac{1}{2}a+\tfrac{1}{2}% \right)+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}},$ $\Re s>0$, $s\neq 1$, $0. ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\Re$: real part, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Keywords: infinite series, series representation Source: Srivastava and Choi (2001, (5), p. 89) Referenced by: (25.11.35), §25.11(x) Permalink: http://dlmf.nist.gov/25.11.E8 Encodings: TeX, pMML, png See also: Annotations for §25.11(iv), §25.11 and Ch.25
 25.11.9 $\zeta\left(1-s,a\right)=\frac{2\Gamma\left(s\right)}{(2\pi)^{s}}\*\sum_{n=1}^{% \infty}\frac{1}{n^{s}}\cos\left(\tfrac{1}{2}\pi s-2n\pi a\right),$ $\Re s>1$, $0.
 25.11.10 $\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{{\left(s\right)_{n}}}{n!}\zeta% \left(n+s\right)(1-a)^{n},$ $s\neq 1$, $|a-1|<1$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, ${\left(\NVar{a}\right)_{\NVar{n}}}$: Pochhammer’s symbol (or shifted factorial), $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $!$: factorial (as in $n!$), $n$: nonnegative integer, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable Keywords: infinite series, series representation Source: Derivable using Taylor’s theorem (1.10.1) with $z=a$, $z_{0}=1$, (25.11.17), (5.2.4), (25.11.2). Referenced by: §25.11(iv), §25.11(iv), §25.11(iv), 1st Erratum 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 $\sum_{n=0}^{\infty}\frac{\Gamma\left(n+s\right)}{n!\Gamma\left(s\right)}\zeta% \left(n+s\right)(1-a)^{n}$ more concisely as $\sum_{n=0}^{\infty}\frac{{\left(s\right)_{n}}}{n!}\zeta\left(n+s\right)(1-a)^{n}$ using the Pochhammer symbol. See also: Annotations for §25.11(iv), §25.11 and Ch.25

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 $\zeta\left(s,\tfrac{1}{2}\right)=(2^{s}-1)\zeta\left(s\right),$ $s\neq 1$. ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function and $s$: complex variable Keywords: special value Source: Derivable using (25.11.1), (25.2.2). 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 and Ch.25
 25.11.12 $\zeta\left(n+1,a\right)=\frac{(-1)^{n+1}{\psi^{(n)}}\left(a\right)}{n!},$ $n=1,2,3,\dots$. ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\psi\left(\NVar{z}\right)$: psi (or digamma) function, $!$: factorial (as in $n!$), $n$: nonnegative integer and $a$: real or complex parameter Keywords: special value Source: Erdélyi et al. (1953a, (9), p. 45) Permalink: http://dlmf.nist.gov/25.11.E12 Encodings: TeX, pMML, png See also: Annotations for §25.11(v), §25.11 and Ch.25
 25.11.13 $\zeta\left(0,a\right)=\tfrac{1}{2}-a.$ ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function and $a$: real or complex parameter Keywords: special value Source: Apostol (1976, p. 268) Permalink: http://dlmf.nist.gov/25.11.E13 Encodings: TeX, pMML, png See also: Annotations for §25.11(v), §25.11 and Ch.25
 25.11.14 $\zeta\left(-n,a\right)=-\frac{B_{n+1}\left(a\right)}{n+1},$ $n=0,1,2,\dots$. ⓘ Symbols: $B_{\NVar{n}}\left(\NVar{x}\right)$: Bernoulli polynomials, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $n$: nonnegative integer and $a$: real or complex parameter Keywords: special value Source: Apostol (1976, (17), p. 264) Permalink: http://dlmf.nist.gov/25.11.E14 Encodings: TeX, pMML, png See also: Annotations for §25.11(v), §25.11 and Ch.25
 25.11.15 $\zeta\left(s,ka\right)=k^{-s}\*\sum_{n=0}^{k-1}\zeta\left(s,a+\frac{n}{k}% \right),$ $s\neq 1$, $k=1,2,3,\dots$. ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $k$: nonnegative integer, $m$: nonnegative integer, $n$: nonnegative integer, $a$: real or complex parameter and $s$: complex variable Keywords: special value Source: Derivable by starting with (25.11.1), replacing $a\mapsto ka$, pulling out a factor of $k^{-s}$ and performing a change of sum index $n=km+l$ with $m\geq 0$ and $0\leq l\leq k-1$, which through Euclidean division converts the single sum into a double sum of the correct form. Referenced by: (25.11.24), §25.11(v), §25.11(vi) Permalink: http://dlmf.nist.gov/25.11.E15 Encodings: TeX, pMML, png See also: Annotations for §25.11(v), §25.11 and Ch.25
 25.11.16 $\zeta\left(1-s,\frac{h}{k}\right)=\frac{2\Gamma\left(s\right)}{(2\pi k)^{s}}\*% \sum_{r=1}^{k}\cos\left(\frac{\pi s}{2}-\frac{2\pi rh}{k}\right)\zeta\left(s,% \frac{r}{k}\right),$ $s\neq 0,1$; $h,k$ integers, $1\leq h\leq k$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$: cosine function, $k$: nonnegative integer and $s$: complex variable Keywords: special value Source: Apostol (1976, (14), p. 261) Referenced by: §25.11(v) Permalink: http://dlmf.nist.gov/25.11.E16 Encodings: TeX, pMML, png See also: Annotations for §25.11(v), §25.11 and Ch.25

## §25.11(vi) Derivatives

### $a$-Derivative

 25.11.17 $\frac{\partial}{\partial a}\zeta\left(s,a\right)=-s\zeta\left(s+1,a\right),$ $s\neq 0,1$; $\Re a>0$. ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$: partial derivative of $f$ with respect to $x$, $\partial\NVar{x}$: partial differential of $x$, $\Re$: real part, $a$: real or complex parameter and $s$: complex variable Keywords: derivative Source: Derivable by differentiating (25.11.1) with respect to $a$. Referenced by: (25.11.10), §25.11(iv), §25.11(vi) Permalink: http://dlmf.nist.gov/25.11.E17 Encodings: TeX, pMML, png See also: Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

### $s$-Derivatives

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

 25.11.18 $\zeta'\left(0,a\right)=\ln\Gamma\left(a\right)-\tfrac{1}{2}\ln\left(2\pi\right),$ $a>0$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$: principal branch of logarithm function and $a$: real or complex parameter Keywords: derivative Source: Erdélyi et al. (1953a, (10), p. 26) Referenced by: §25.11(vi), §25.11(vi) Permalink: http://dlmf.nist.gov/25.11.E18 Encodings: TeX, pMML, png See also: Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25
 25.11.19 $\zeta'\left(s,a\right)=-\frac{\ln a}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}% \right)-\frac{a^{1-s}}{(s-1)^{2}}+\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(% \widetilde{B}_{2}\left(x\right)-B_{2})\ln\left(x+a\right)}{(x+a)^{s+2}}\mathrm% {d}x-\frac{(2s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x\right)-B_% {2}}{(x+a)^{s+2}}\mathrm{d}x,$ $\Re s>-1$, $s\neq 1$, $a>0$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\ln\NVar{z}$: principal branch of logarithm function, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$: periodic Bernoulli functions, $\Re$: real part, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $k$: integer Keywords: derivative Source: Apostol (1985a, fourth equation, p. 231); with $k=1$, $\varphi_{2}(x)=\frac{1}{2}(\widetilde{B}_{2}\left(x\right)-B_{2})$ Referenced by: Erratum 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 $\widetilde{B}_{2}\left(x\right)$. The correct function is $\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{2}$. Reported 2016-06-27 by Gergő Nemes See also: Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25
 25.11.20 $(-1)^{k}{\zeta^{(k)}}\left(s,a\right)=\frac{(\ln a)^{k}}{a^{s}}\left(\frac{1}{% 2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum_{r=0}^{k-1}\frac{(\ln a)^{r}}{r!(s-1)^{k% -r+1}}-\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)% -B_{2})(\ln\left(x+a\right))^{k}}{(x+a)^{s+2}}\mathrm{d}x+\frac{k(2s+1)}{2}% \int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a% \right))^{k-1}}{(x+a)^{s+2}}\mathrm{d}x-\frac{k(k-1)}{2}\int_{0}^{\infty}\frac% {(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a\right))^{k-2}}{(x+a)^{s+% 2}}\mathrm{d}x,$ $\Re s>-1$, $s\neq 1$, $a>0$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\mathrm{d}\NVar{x}$: differential of $x$, $!$: factorial (as in $n!$), $\int$: integral, $\ln\NVar{z}$: principal branch of logarithm function, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$: periodic Bernoulli functions, $\Re$: real part, $x$: real variable, $a$: real or complex parameter, $s$: complex variable and $k$: integer Keywords: derivative Source: Apostol (1985a, fourth equation, p. 231); with $\varphi_{2}(x)=\frac{1}{2}(\widetilde{B}_{2}\left(x\right)-B_{2})$ Referenced by: Erratum 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 $\widetilde{B}_{2}\left(x\right)$. The correct function is $\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{2}$. Reported 2016-06-27 by Gergő Nemes See also: Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25
 25.11.21 $\zeta'\left(1-2n,\frac{h}{k}\right)=\frac{(\psi\left(2n\right)-\ln\left(2\pi k% \right))B_{2n}\left(h/k\right)}{2n}-\frac{(\psi\left(2n\right)-\ln\left(2\pi% \right))B_{2n}}{2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}% \sin\left(\frac{2\pi rh}{k}\right){\psi^{(2n-1)}}\left(\frac{r}{k}\right)+% \frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\cos\left(\frac{2% \pi rh}{k}\right)\zeta'\left(2n,\frac{r}{k}\right)+\frac{\zeta'\left(1-2n% \right)}{k^{2n}},$

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

 25.11.22 $\zeta'\left(1-2n,\tfrac{1}{2}\right)=-\frac{B_{2n}\ln 2}{n\cdot 4^{n}}-\frac{(% 2^{2n-1}-1)\zeta'\left(1-2n\right)}{2^{2n-1}},$ $n=1,2,3,\dots$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\ln\NVar{z}$: principal branch of logarithm function and $n$: nonnegative integer Keywords: derivative Source: Miller and Adamchik (1998, (17), p. 205) Permalink: http://dlmf.nist.gov/25.11.E22 Encodings: TeX, pMML, png See also: Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25
 25.11.23 $\zeta'\left(1-2n,\tfrac{1}{3}\right)=-\frac{\pi(9^{n}-1)B_{2n}}{8n\sqrt{3}(3^{% 2n-1}-1)}-\frac{B_{2n}\ln 3}{4n\cdot 3^{2n-1}}-\frac{(-1)^{n}{\psi^{(2n-1)}}% \left(\frac{1}{3}\right)}{2\sqrt{3}(6\pi)^{2n-1}}-\frac{\left(3^{2n-1}-1\right% )\zeta'\left(1-2n\right)}{2\cdot 3^{2n-1}},$ $n=1,2,3,\dots$.
 25.11.24 $\sum_{r=1}^{k-1}\zeta'\left(s,\frac{r}{k}\right)=(k^{s}-1)\zeta'\left(s\right)% +k^{s}\zeta\left(s\right)\ln k,$ $s\neq 1$, $k=1,2,3,\dots$. ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $\ln\NVar{z}$: principal branch of logarithm function, $a$: real or complex parameter, $s$: complex variable and $k$: integer Keywords: derivative Source: Derivable from (25.11.15) with $a=1/k$, multiplying by $k^{s}$, differentiating with respect to $s$, and using (25.11.2). Referenced by: §25.11(vi), §25.11(vi) Permalink: http://dlmf.nist.gov/25.11.E24 Encodings: TeX, pMML, png See also: Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

## §25.11(vii) Integral Representations

 25.11.25 $\displaystyle\zeta\left(s,a\right)$ $\displaystyle=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-% ax}}{1-e^{-x}}\mathrm{d}x,$ $\Re s>1$, $\Re a>0$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\Re$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: improper integral, integral representation Sources: Srivastava and Choi (2001, (2), p. 89); Apostol (1976, p. 252) Referenced by: (25.11.27), (25.11.30), (25.11.35), §25.11(vii), §25.11(viii), §25.11(x) Permalink: http://dlmf.nist.gov/25.11.E25 Encodings: TeX, pMML, png See also: Annotations for §25.11(vii), §25.11 and Ch.25 25.11.26 $\displaystyle\zeta\left(s,a\right)$ $\displaystyle=-s\int_{-a}^{\infty}\frac{x-\left\lfloor x\right\rfloor-\frac{1}% {2}}{(x+a)^{s+1}}\mathrm{d}x,$ $-1<\Re s<0$, $0.
 25.11.27 $\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\frac{1}{\Gamma% \left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2% }\right)\frac{x^{s-1}}{e^{ax}}\mathrm{d}x,$ $\Re s>-1$, $s\neq 1$, $\Re a>0$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $\Re$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: improper integral, integral representation Source: Derivable from (25.11.25) by first rewriting $(1-{\mathrm{e}^{-x}})={\mathrm{e}^{-x}}({\mathrm{e}^{x}}-1)$, then replacing $x^{s-1}\mapsto x^{s-1}(1+{\mathrm{e}^{-x}}-{\mathrm{e}^{-x}})$, using the identity ${\mathrm{e}^{-x}}=(1-{\mathrm{e}^{-x}})/({\mathrm{e}^{x}}-1)$, and replacing $({\mathrm{e}^{x}}-1)^{-1}\mapsto(({\mathrm{e}^{x}}-1)^{-1}-1/x+1/2-1/2+1/x)$ in the resulting integrand with (5.2.1), (5.5.1). Referenced by: (25.11.28), §25.11(vii) Permalink: http://dlmf.nist.gov/25.11.E27 Encodings: TeX, pMML, png See also: Annotations for §25.11(vii), §25.11 and Ch.25
 25.11.28 $\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\sum_{k=1}^{n}% \frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}+\frac{1}{\Gamma\left(s% \right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_% {k=1}^{n}\frac{B_{2k}}{(2k)!}x^{2k-1}\right)x^{s-1}e^{-ax}\mathrm{d}x,$ $\Re s>-(2n+1)$, $s\neq 1$, $\Re a>0$. ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $O\left(\NVar{x}\right)$: order not exceeding, $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$: 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 natural logarithm, $!$: 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 Keywords: improper integral, integral representation Source: Derivable from (25.11.27) by adding and subtracting $\sum_{k=1}^{n}B_{2k}x^{2k-1}/(2k)!$ in the integrand, using (5.2.1), (5.2.5), and recognizing that $({\mathrm{e}^{x}}-1)^{-1}-x^{-1}+1/2-\sum_{k=1}^{n}B_{2k}x^{2k-1}/(2k)!=O\left% (x^{2n+1}\right)$ as $x\to 0$, demonstrating the region of convergence. Referenced by: §25.11(vii), 1st Erratum 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 $\sum_{k=1}^{n}\frac{\Gamma\left(s+2k-1\right)}{\Gamma\left(s\right)}\frac{B_{2% k}}{(2k)!}a^{-2k-s+1}$ more concisely as $\sum_{k=1}^{n}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}$ using the Pochhammer symbol. See also: Annotations for §25.11(vii), §25.11 and Ch.25
 25.11.29 $\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+2\int_{0}^{\infty}% \frac{\sin\left(s\operatorname{arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{% s/2}(e^{2\pi x}-1)}\mathrm{d}x,$ $s\neq 1$, $\Re a>0$.
 25.11.30 $\zeta\left(s,a\right)=\frac{\Gamma\left(1-s\right)}{2\pi i}\int_{-\infty}^{(0+% )}\frac{e^{az}z^{s-1}}{1-e^{z}}\mathrm{d}z,$ $s\neq 1$, $\Re a>0$, ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $\Re$: real part, $a$: real or complex parameter, $s$: complex variable and $z$: complex variable Keywords: contour integral, integral representation Source: This contour integral uses Hankel’s contour Figure 5.9.1. Assume $\Re{s}>1$, collapse the integration path onto the negative real axis, apply (25.11.25), followed by analytic continuation for all $s\neq 1$, $\Re{a}>0$, since the integrand is analytic in $z$, the convergence at the ends of the path is exponential for all $s\neq-1$ and the Hankel contour can be kept well clear of the singularity at the origin in the $z$-plane. Referenced by: §25.11(i), §25.11(vii) Permalink: http://dlmf.nist.gov/25.11.E30 Encodings: TeX, pMML, png See also: Annotations for §25.11(vii), §25.11 and Ch.25

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 $\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{2\cosh x}% \mathrm{d}x=4^{-s}\left(\zeta\left(s,\tfrac{1}{4}+\tfrac{1}{4}a\right)-\zeta% \left(s,\tfrac{3}{4}+\tfrac{1}{4}a\right)\right),$ $\Re s>0$, $\Re a>-1$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\cosh\NVar{z}$: hyperbolic cosine function, $\int$: integral, $\Re$: real part, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: improper integral, integral representation Source: The improper integral can be evaluated by using $2\cosh x={\mathrm{e}^{x}}(1+{\mathrm{e}^{-2x}})$, changing variables $2x=t$, and then applying (25.11.35). Referenced by: §25.11(viii) Permalink: http://dlmf.nist.gov/25.11.E31 Encodings: TeX, pMML, png See also: Annotations for §25.11(viii), §25.11 and Ch.25
 25.11.32 $\int_{0}^{a}x^{n}\psi\left(x\right)\mathrm{d}x=(-1)^{n-1}\zeta'\left(-n\right)% +(-1)^{n}h(n)\frac{B_{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0.0pt}{}% {n}{k}h(k)\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{0% .0pt}{}{n}{k}\zeta'\left(-k,a\right)a^{n-k},$ $n=1,2,\dots$, $\Re a>0$,

where

 25.11.33 $h(n)=\sum_{k=1}^{n}k^{-1}.$ ⓘ Symbols: $k$: nonnegative integer, $n$: nonnegative integer and $h(n)$: sum Keywords: harmonic number Source: Knuth (1968, Section 1.2.7, p. 73) 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 and Ch.25
 25.11.34 $n\int_{0}^{a}\zeta'\left(1-n,x\right)\mathrm{d}x=\zeta'\left(-n,a\right)-\zeta% '\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a\right)}{n(n+1)},$ $n=1,2,\dots$, $\Re a>0$.

## §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 $\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}}=\frac{1}{\Gamma\left(s\right)}% \int_{0}^{\infty}\frac{x^{s-1}e^{-ax}}{1+e^{-x}}\mathrm{d}x=2^{-s}\left(\zeta% \left(s,\tfrac{1}{2}a\right)-\zeta\left(s,\tfrac{1}{2}(1+a)\right)\right),$ $\Re a>0$, $\Re s>0$; or $\Re a=0$, $\Im a\neq 0$, $0<\Re s<1$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\Im$: imaginary part, $\int$: integral, $\Re$: real part, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $s$: complex variable Keywords: improper integral, infinite series, integral representation, series representation Source: The infinite series connection to the difference of Hurwitz zeta functions is a restatement of (25.11.8). The integral representation is derivable from the difference of Hurwitz zeta functions by substituting (25.11.25) in each, making a change of variables $x=2t$, factoring, and expressing $1-{\mathrm{e}^{-2t}}=(1+{\mathrm{e}^{-t}})(1-{\mathrm{e}^{-t}})$. Referenced by: (25.11.31), §25.11(x), §25.11(x) Permalink: http://dlmf.nist.gov/25.11.E35 Encodings: TeX, pMML, png See also: Annotations for §25.11(x), §25.11 and Ch.25

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

 25.11.36 $L\left(s,\chi\right)=\sum_{n=1}^{\infty}\frac{\chi(n)}{n^{s}}=k^{-s}\sum_{r=1}% ^{k-1}\chi(r)\zeta\left(s,\frac{r}{k}\right),$ $\Re s>1$, ⓘ Symbols: $L\left(\NVar{s},\NVar{\chi}\right)$: Dirichlet $L$-function, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\Re$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $s$: complex variable and $\chi(n)$: Dirichlet character Keywords: infinite series, series representation Source: Apostol (1976, p. 249) Referenced by: §25.11(x), 3rd Erratum Permalink: http://dlmf.nist.gov/25.11.E36 Encodings: TeX, pMML, png Clarification (effective with 1.0.23): The Dirichlet $L$-function was inserted on the left-hand side. The upper-index of the finite sum which originally was $k$, was replaced with $k-1$ since $\chi(k)=0$. See also: Annotations for §25.11(x), §25.11 and Ch.25

where $\chi(n)$ is a Dirichlet character $\pmod{k}$27.8). Compare (25.15.1) and (25.15.3).

## §25.11(xi) Sums

 25.11.37 $\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}\zeta\left(nk,a\right)=-n\ln\Gamma\left(a% \right)+\ln\left(\prod_{j=0}^{n-1}\Gamma\left(a-e^{(2j+1)\pi i/n}\right)\right),$ $n=2,3,4,\dots$, $\Re a\geq 1$. ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\ln\NVar{z}$: principal branch of logarithm function, $\operatorname{ph}$: phase, $\Re$: real part, $k$: nonnegative integer, $n$: nonnegative integer and $a$: real or complex parameter Keywords: infinite series, series representation Source: Adamchik and Srivastava (1998, (2.12), p. 136); with the choice $\operatorname{ph}\left(-1\right)=\pi$, which is reasonable since the gamma function is single valued Permalink: http://dlmf.nist.gov/25.11.E37 Encodings: TeX, pMML, png See also: Annotations for §25.11(xi), §25.11 and Ch.25
 25.11.38 $\sum_{k=1}^{\infty}\genfrac{(}{)}{0.0pt}{}{n+k}{k}\zeta\left(n+k+1,a\right)z^{% k}=\frac{(-1)^{n}}{n!}\left({\psi^{(n)}}\left(a\right)-{\psi^{(n)}}\left(a-z% \right)\right),$ $n=1,2,3,\dots$, $\Re a>0$, $|z|<|a|$.
 25.11.39 $\sum_{k=2}^{\infty}\frac{k}{2^{k}}\zeta\left(k+1,\tfrac{3}{4}\right)=8G,$ ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $k$: nonnegative integer and $G$: Catalan’s constant Keywords: Catalan’s constant, infinite series Source: Adamchik and Srivastava (1998, (2.30), p. 138) Permalink: http://dlmf.nist.gov/25.11.E39 Encodings: TeX, pMML, png See also: Annotations for §25.11(xi), §25.11 and Ch.25

where $G$ is Catalan’s constant:

 25.11.40 $G\equiv\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}=0.91596\;55941\;772\dots.$ ⓘ Symbols: $\equiv$: equals by definition, $n$: nonnegative integer and $G$: Catalan’s constant Keywords: Catalan’s constant, definition Source: Adamchik and Srivastava (1998, (2.32), p. 139) 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), §25.11 and Ch.25

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 $\zeta\left(s,a+1\right)=\zeta\left(s\right)-s\zeta\left(s+1\right)a+O\left(a^{% 2}\right).$ ⓘ Symbols: $O\left(\NVar{x}\right)$: order not exceeding, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$: Riemann zeta function, $a$: real or complex parameter and $s$: complex variable Keywords: asymptotic approximation Source: Apostol (1952, (15), p. 6) Referenced by: §25.11(xii) Permalink: http://dlmf.nist.gov/25.11.E41 Encodings: TeX, pMML, png See also: Annotations for §25.11(xii), §25.11 and Ch.25

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

 25.11.42 $\zeta\left(s,\alpha+i\beta\right)\to 0,$ ⓘ Symbols: $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\mathrm{i}$: imaginary unit and $s$: complex variable Keywords: asymptotic approximation Source: Apostol (1952, (16), p. 6) Referenced by: §25.11(xii) Permalink: http://dlmf.nist.gov/25.11.E42 Encodings: TeX, pMML, png See also: Annotations for §25.11(xii), §25.11 and Ch.25

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

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

 25.11.43 $\zeta\left(s,a\right)-\frac{a^{1-s}}{s-1}-\frac{1}{2}a^{-s}\sim\sum_{k=1}^{% \infty}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}.$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\Gamma\left(\NVar{z}\right)$: gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$: 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 Keywords: asymptotic approximation Source: Paris (2005b, (1.3), (1.4), p. 298) Referenced by: §25.11(xii), §25.11(xii), 1st Erratum 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 $\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}\frac{\Gamma\left(s+2k-1\right)}{\Gamma% \left(s\right)}a^{1-s-2k}$ more concisely as $\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}$ using the Pochhammer symbol. See also: Annotations for §25.11(xii), §25.11 and Ch.25

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

 25.11.44 $\zeta'\left(-1,a\right)-\frac{1}{12}+\frac{1}{4}a^{2}-\left(\frac{1}{12}-\frac% {1}{2}a+\frac{1}{2}a^{2}\right)\ln a\sim-\sum_{k=1}^{\infty}\frac{B_{2k+2}}{(2% k+2)(2k+1)2k}a^{-2k},$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\sim$: Poincaré asymptotic expansion, $\ln\NVar{z}$: principal branch of logarithm function, $k$: nonnegative integer and $a$: real or complex parameter Keywords: asymptotic approximation Source: Elizalde (1986, (18), p. 349) Referenced by: §25.11(xii) Permalink: http://dlmf.nist.gov/25.11.E44 Encodings: TeX, pMML, png See also: Annotations for §25.11(xii), §25.11 and Ch.25

and

 25.11.45 $\zeta'\left(-2,a\right)-\frac{1}{12}a+\frac{1}{9}a^{3}-\left(\frac{1}{6}a-% \frac{1}{2}a^{2}+\frac{1}{3}a^{3}\right)\ln a\sim\sum_{k=1}^{\infty}\frac{2B_{% 2k+2}}{(2k+2)(2k+1)2k(2k-1)}a^{-(2k-1)}.$ ⓘ Symbols: $B_{\NVar{n}}$: Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$: Hurwitz zeta function, $\sim$: Poincaré asymptotic expansion, $\ln\NVar{z}$: principal branch of logarithm function, $k$: nonnegative integer and $a$: real or complex parameter Keywords: asymptotic approximation Source: Elizalde (1986, (19), p. 350) Referenced by: §25.11(xii) Permalink: http://dlmf.nist.gov/25.11.E45 Encodings: TeX, pMML, png See also: Annotations for §25.11(xii), §25.11 and Ch.25

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

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