§25.11 Hurwitz Zeta Function

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Keywords:: Hurwitz zeta function, analytic properties
Notes:: See Apostol (1976, Chapter 12), Srivastava and Choi (2001), Miller and Adamchik (1998), and Adamchik and Srivastava (1998).
Referenced by:: §25.14(i)
Permalink:: http://dlmf.nist.gov/25.11
See also:: Annotations for Ch.25

§25.11(i) Definition
§25.11(ii) Graphics
§25.11(iii) Representations by the Euler–Maclaurin Formula
§25.11(iv) Series Representations
§25.11(v) Special Values
§25.11(vi) Derivatives
§25.11(vii) Integral Representations
§25.11(viii) Further Integral Representations
§25.11(ix) Integrals
§25.11(x) Further Series Representations
§25.11(xi) Sums
§25.11(xii) -Asymptotic Behavior

§25.11(i) Definition

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Keywords:: Hurwitz zeta function, Riemann zeta function, definition, relations to other functions
Notes:: Analytic properties of $\zeta\left(s,a\right)$ with respect to follow from (25.11.30).
Referenced by:: §8.15
Permalink:: http://dlmf.nist.gov/25.11.i
See also:: Annotations for §25.11 and Ch.25

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

25.11.1 $\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{1}{(n+a)^{s}},$ $\Re s>1$ , $a\neq 0,-1,-2,\dots$ .

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Defines:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function
Symbols:: $\Re$ : real part, : nonnegative integer, : real or complex parameter and : complex variable
Keywords:: definition
Source:: Apostol (1976, p. 251)
Referenced by:: (25.11.11), (25.11.15), (25.11.17), (25.11.2), (25.11.3), (25.11.4), §25.11(i), (25.14.2), (8.15.2)
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 -plane, its only singularity in $\mathbb{C}$ being a simple pole at with residue 1. As a function of , with ( $\neq 1$ ) 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 $\zeta\left(s,1\right)=\zeta\left(s\right).$

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Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and : complex variable
Keywords:: specialization
Source:: Apostol (1976, p. 255)
Proof sketch:: Derivable from (25.11.1) and (25.2.1).
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 $\zeta\left(s,a\right)=\zeta\left(s,a+1\right)+a^{-s},$

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Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, : nonnegative integer, : real or complex parameter and : complex variable
Keywords:: recurrence
Proof sketch:: Derivable from (25.11.1) by comparing its 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, : nonnegative integer, : nonnegative integer, : real or complex parameter and : complex variable
Keywords:: recurrence
Proof sketch:: 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 with $0<a\leq 1$ .

§25.11(ii) Graphics

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Keywords:: Hurwitz zeta function, graphics
Notes:: These graphics were constructed at NIST.
Permalink:: http://dlmf.nist.gov/25.11.ii
See also:: Annotations for §25.11 and Ch.25

Figure 25.11.1: Hurwitz zeta function $\zeta\left(x,a\right)$ ,

= 0.3, 0.5, 0.8, 1, $-20\leq x\leq 10$ . The curves are almost indistinguishable for

, approximately. Magnify 3D Help

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Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, : real variable and : real or complex parameter
Permalink:: http://dlmf.nist.gov/25.11.F1
Encodings:: pdf, png
See also:: Annotations for §25.11(ii), §25.11 and Ch.25

Figure 25.11.2: Hurwitz zeta function $\zeta\left(x,a\right)$ , $-19.5\leq x\leq 10$ , $0.02\leq a\leq 1$ . Magnify 3D Help

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Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, : real variable and : real or complex parameter
Permalink:: http://dlmf.nist.gov/25.11.F2
Encodings:: Vizualization, pdf, png
See also:: Annotations for §25.11(ii), §25.11 and Ch.25

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

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Keywords:: Hurwitz zeta function, representations by Euler–Maclaurin formula
Permalink:: http://dlmf.nist.gov/25.11.iii
See also:: Annotations for §25.11 and Ch.25

25.11.5 $\zeta\left(s,a\right)=\sum_{n=0}^{N}\frac{1}{(n+a)^{s}}+\frac{(N+a)^{1-s}}{s-1% }-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{(x+a)^{s+1}}\,\mathrm% {d}x,$ $s\neq 1$ , $\Re s>0$ ,

, $N=0,1,2,3,\dots$ .

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Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\left\lfloor\NVar{x}\right\rfloor$ : floor of , $\int$ : integral, $\Re$ : real part, : nonnegative integer, : real variable, : real or complex parameter and : complex variable
Keywords:: Euler–Maclaurin formula, improper integral
Source:: Apostol (1976, (25), p. 269)
Referenced by:: (25.11.7)
Permalink:: http://dlmf.nist.gov/25.11.E5
Encodings:: TeX, pMML, png
See also:: Annotations for §25.11(iii), §25.11 and Ch.25

25.11.6 $\zeta\left(s,a\right)=\frac{1}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}\right)-% \frac{s(s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x\right)-B_{2}}{% (x+a)^{s+2}}\,\mathrm{d}x,$ $s\neq 1$ , $\Re s>-1$ ,

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\Re$ : real part, : real variable, : real or complex parameter and : complex variable
Keywords:: Euler–Maclaurin formula, improper integral
Source:: Apostol (1985a, (25), p. 231)
Referenced by:: Erratum (V1.0.12) for 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 $\zeta\left(s,a\right)=\frac{1}{a^{s}}+\frac{1}{(1+a)^{s}}\left(\frac{1}{2}+% \frac{1+a}{s-1}\right)+\sum_{k=1}^{n}\genfrac{(}{)}{0.0pt}{}{s+2k-2}{2k-1}% \frac{B_{2k}}{2k}\frac{1}{(1+a)^{s+2k-1}}-\genfrac{(}{)}{0.0pt}{}{s+2n}{2n+1}% \int_{1}^{\infty}\frac{\widetilde{B}_{2n+1}\left(x\right)}{(x+a)^{s+2n+1}}\,% \mathrm{d}x,$ $s\neq 1$ ,

, $n=1,2,3,\dots$ , $\Re s>-2n$ .

ⓘ

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 , $\int$ : integral, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\Re$ : real part, : nonnegative integer, : nonnegative integer, : real variable, : real or complex parameter and : complex variable
Keywords:: Euler–Maclaurin formula, improper integral
Proof sketch:: Derivable from (25.11.5) by setting and repeatedly integrating by parts using (1.2.6), (24.2.2), (24.2.4), (24.2.11), (24.2.12), (24.4.34).
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

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Keywords:: Hurwitz zeta function, series representations
Notes:: See Srivastava and Choi (2001) and Apostol (1976, Section 12.7).
Referenced by:: Erratum (V1.0.5) for References
Permalink:: http://dlmf.nist.gov/25.11.iv
Addition (effective with 1.0.5):: The reference to Coffey (2008) has been added at the end of this subsection.
See also:: Annotations for §25.11 and Ch.25

25.11.8 $\zeta\left(s,\tfrac{1}{2}a\right)=\zeta\left(s,\tfrac{1}{2}a+\tfrac{1}{2}% \right)+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a)^{s}},$ $\Re s>0$ , $s\neq 1$ , $0<a\leq 1$ .

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Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\Re$ : real part, : nonnegative integer, : real or complex parameter and : 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 $\zeta\left(1-s,a\right)=\frac{2\Gamma\left(s\right)}{(2\pi)^{s}}\*\sum_{n=1}^{% \infty}\frac{1}{n^{s}}\cos\left(\tfrac{1}{2}\pi s-2n\pi a\right),$ $\Re s>0$ if

; $\Re s>1$ if

ⓘ

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, $\Re$ : real part, : nonnegative integer, : real or complex parameter and : complex variable
Keywords:: infinite series, series representation
Source:: Apostol (1976, (9), (10), p. 257)
Referenced by:: (25.13.3), Erratum (V1.1.4) for Equation (25.11.9)
Permalink:: http://dlmf.nist.gov/25.11.E9
Encodings:: TeX, pMML, png
Clarification (effective with 1.1.4):: The constraint which originally read “ $\Re s>1$ , $0<a\leq 1$ ” has been extended to be “ $\Re s>0$ if ; $\Re s>1$ if ”.
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.11(iv), §25.11 and Ch.25

25.11.10 $\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{{\left(s\right)_{n}}}{n!}\zeta% \left(n+s\right)(1-a)^{n},$ $s\neq 1$ ,

ⓘ

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 ), : nonnegative integer, : real or complex parameter, : complex variable and : complex variable
Keywords:: infinite series, series representation
Proof sketch:: Derivable using Taylor’s theorem (1.10.1) with , $z_{0}=1$ , (25.11.17), (5.2.4), (25.11.2).
Referenced by:: §25.11(iv), §25.11(iv), Erratum (V1.0.9) for Chapters 7, 25
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

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Keywords:: Hurwitz zeta function, special values
Permalink:: http://dlmf.nist.gov/25.11.v
See also:: Annotations for §25.11 and Ch.25

Throughout this subsection $\Re a>0$ .

25.11.11 $\zeta\left(s,\tfrac{1}{2}\right)=(2^{s}-1)\zeta\left(s\right),$ $s\neq 1$ .

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Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function and : complex variable
Keywords:: special value
Proof sketch:: Derivable using (25.11.1), (25.2.2).
Referenced by:: §25.11(iv)
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 ), : nonnegative integer and : real or complex parameter
Keywords:: special value
Source:: Erdélyi et al. (1953a, (1.11.8), p. 29)
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 : 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, : nonnegative integer and : 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, : nonnegative integer, : nonnegative integer, : nonnegative integer, : real or complex parameter and : complex variable
Keywords:: special value
Proof sketch:: 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 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)
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 $\zeta\left(1-s,\frac{h}{k}\right)=\frac{2\Gamma\left(s\right)}{(2\pi k)^{s}}\*% \sum_{r=1}^{k}\cos\left(\frac{\pi s}{2}-\frac{2\pi rh}{k}\right)\zeta\left(s,% \frac{r}{k}\right),$ $s\neq 0,1$ ;

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, : nonnegative integer and : complex variable
Keywords:: special value
Source:: Apostol (1976, (14), p. 261)
Referenced by:: (25.11.21)
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

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Keywords:: Hurwitz zeta function, derivatives
Notes:: See Apostol (1985a, p. 231) and Miller and Adamchik (1998).
Permalink:: http://dlmf.nist.gov/25.11.vi
See also:: Annotations for §25.11 and Ch.25

-Derivative

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See also:: Annotations for §25.11(vi), §25.11 and Ch.25

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 with respect to , $\,\partial\NVar{x}$ : partial differential of , $\Re$ : real part, : real or complex parameter and : complex variable
Keywords:: derivative
Proof sketch:: Derivable by differentiating (25.11.1) with respect to .
Referenced by:: (25.11.10)
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

-Derivatives

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See also:: Annotations for §25.11(vi), §25.11 and Ch.25

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

25.11.18 $\zeta'\left(0,a\right)=\ln\Gamma\left(a\right)-\tfrac{1}{2}\ln\left(2\pi\right),$

ⓘ

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 : real or complex parameter
Keywords:: derivative
Source:: Erdélyi et al. (1953a, (1.10.10), p. 26)
Referenced by:: §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 $\zeta'\left(s,a\right)=-\frac{\ln a}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}% \right)-\frac{a^{1-s}}{(s-1)^{2}}+\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(% \widetilde{B}_{2}\left(x\right)-B_{2})\ln\left(x+a\right)}{(x+a)^{s+2}}\,% \mathrm{d}x-\frac{(2s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x% \right)-B_{2}}{(x+a)^{s+2}}\,\mathrm{d}x,$ $\Re s>-1$ , $s\neq 1$ ,

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\Re$ : real part, : real variable, : real or complex parameter, : complex variable and : 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 (V1.0.12) for 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 $(-1)^{k}{\zeta}^{(k)}\left(s,a\right)=\frac{(\ln a)^{k}}{a^{s}}\left(\frac{1}{% 2}+\frac{a}{s-1}\right)+k!a^{1-s}\sum_{r=0}^{k-1}\frac{(\ln a)^{r}}{r!(s-1)^{k% -r+1}}-\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)% -B_{2})(\ln\left(x+a\right))^{k}}{(x+a)^{s+2}}\,\mathrm{d}x+\frac{k(2s+1)}{2}% \int_{0}^{\infty}\frac{(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a% \right))^{k-1}}{(x+a)^{s+2}}\,\mathrm{d}x-\frac{k(k-1)}{2}\int_{0}^{\infty}% \frac{(\widetilde{B}_{2}\left(x\right)-B_{2})(\ln\left(x+a\right))^{k-2}}{(x+a% )^{s+2}}\,\mathrm{d}x,$ $\Re s>-1$ , $s\neq 1$ ,

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , : factorial (as in ), $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\widetilde{B}_{\NVar{n}}\left(\NVar{x}\right)$ : periodic Bernoulli functions, $\Re$ : real part, : real variable, : real or complex parameter, : complex variable and : 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 (V1.0.12) for 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 $\zeta'\left(1-2n,\frac{h}{k}\right)=\frac{(\psi\left(2n\right)-\ln\left(2\pi k% \right))B_{2n}\left(h/k\right)}{2n}-\frac{(\psi\left(2n\right)-\ln\left(2\pi% \right))B_{2n}}{2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}% \sin\left(\frac{2\pi rh}{k}\right){\psi}^{(2n-1)}\left(\frac{r}{k}\right)+% \frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=1}^{k-1}\cos\left(\frac{2% \pi rh}{k}\right)\zeta'\left(2n,\frac{r}{k}\right)+\frac{\zeta'\left(1-2n% \right)}{k^{2n}},$

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : factorial (as in ), $\ln\NVar{z}$ : principal branch of logarithm function, $\sin\NVar{z}$ : sine function, : nonnegative integer, : integer and : integer
Keywords:: derivative
Source:: Miller and Adamchik (1998, (5), p. 203)
Notes:: According to the source, (25.11.21) is valid only for $1\leq h<k$ . Their derivation, however, is based on (25.11.16) which is true when as well. Alternatively, one can substitute into (25.11.21) and simplify via (25.11.24), (25.6.2) and (25.6.15) confirming that (25.11.21) holds also in the case that .
Referenced by:: (25.11.21)
Permalink:: http://dlmf.nist.gov/25.11.E21
Encodings:: TeX, pMML, png
See also:: Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

where 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 : 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 $\zeta'\left(1-2n,\tfrac{1}{3}\right)=-\frac{\pi(9^{n}-1)B_{2n}}{8n\sqrt{3}(3^{% 2n-1}-1)}-\frac{B_{2n}\ln 3}{4n\cdot 3^{2n-1}}-\frac{(-1)^{n}{\psi}^{(2n-1)}% \left(\frac{1}{3}\right)}{2\sqrt{3}(6\pi)^{2n-1}}-\frac{\left(3^{2n-1}-1\right% )\zeta'\left(1-2n\right)}{2\cdot 3^{2n-1}},$ $n=1,2,3,\dots$ .

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $\ln\NVar{z}$ : principal branch of logarithm function and : nonnegative integer
Keywords:: derivative
Source:: Miller and Adamchik (1998, (18), p. 205)
Permalink:: http://dlmf.nist.gov/25.11.E23
Encodings:: TeX, pMML, png
See also:: Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

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, : real or complex parameter, : complex variable and : integer
Keywords:: derivative
Proof sketch:: Derivable from (25.11.15) with , multiplying by $k^{s}$ , differentiating with respect to , and using (25.11.2).
Referenced by:: (25.11.21), §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

ⓘ

Keywords:: Hurwitz zeta function, integral representations
Notes:: See Apostol (1976, Section 12.3).
Permalink:: http://dlmf.nist.gov/25.11.vii
Clarification (effective with 1.2.1):: Just below (25.11.30), we added “(see Figure 5.9.1)”.
See also:: Annotations for §25.11 and Ch.25

25.11.25 $\zeta\left(s,a\right)=\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{% s-1}e^{-ax}}{1-e^{-x}}\,\mathrm{d}x,$ $\Re s>1$ , $\Re a>0$ .

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, : real variable, : real or complex parameter and : complex variable
Keywords:: improper integral, integral representation
Sources:: Srivastava and Choi (2001, (2), p. 89); Apostol (1976, (5), p. 251)
Referenced by:: (25.11.27), (25.11.30), (25.11.35), §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 $\zeta\left(s,a\right)=-s\int_{-a}^{\infty}\frac{x-\left\lfloor x\right\rfloor-% \frac{1}{2}}{(x+a)^{s+1}}\,\mathrm{d}x,$ $-1<\Re s<0$ , $0<a\leq 1$ .

ⓘ

Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\left\lfloor\NVar{x}\right\rfloor$ : floor of , $\int$ : integral, $\Re$ : real part, : real variable, : real or complex parameter and : complex variable
Keywords:: improper integral, integral representation
Source:: Berndt (1972, (5.3), p. 156)
Permalink:: http://dlmf.nist.gov/25.11.E26
Encodings:: TeX, pMML, png
See also:: Annotations for §25.11(vii), §25.11 and Ch.25

25.11.27 $\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+\frac{1}{\Gamma% \left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2% }\right)x^{s-1}e^{-ax}\,\mathrm{d}x,$ $\Re s>-1$ , $s\neq 1$ , $\Re a>0$ .

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\Re$ : real part, : real variable, : real or complex parameter and : complex variable
Keywords:: improper integral, integral representation
Proof sketch:: 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)
Permalink:: http://dlmf.nist.gov/25.11.E27
Encodings:: TeX, pMML, png
Update (effective with 1.1.9):: The fraction in the integrand $\frac{x^{s-1}}{e^{ax}}$ was rewritten to read $x^{s-1}e^{-ax}$ .
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 , $\mathrm{e}$ : base of natural logarithm, : factorial (as in ), $\int$ : integral, $\Re$ : real part, : nonnegative integer, : nonnegative integer, : real variable, : real or complex parameter and : complex variable
Keywords:: improper integral, integral representation
Proof sketch:: 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:: Erratum (V1.0.9) for Chapters 7, 25
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 $\zeta\left(s,a\right)=\frac{1}{2}a^{-s}+\frac{a^{1-s}}{s-1}+2\int_{0}^{\infty}% \frac{\sin\left(s\operatorname{arctan}\left(x/a\right)\right)}{(a^{2}+x^{2})^{% s/2}(e^{2\pi x}-1)}\,\mathrm{d}x,$ $s\neq 1$ , $\Re a>0$ .

ⓘ

Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\Re$ : real part, $\sin\NVar{z}$ : sine function, : real variable, : real or complex parameter and : complex variable
Keywords:: improper integral, integral representation
Source:: Erdélyi et al. (1953a, (1.10.7), p. 26)
Permalink:: http://dlmf.nist.gov/25.11.E29
Encodings:: TeX, pMML, png
See also:: Annotations for §25.11(vii), §25.11 and Ch.25

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 , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, : real or complex parameter, : complex variable and : complex variable
Keywords:: contour integral, integral representation
Source:: Erdélyi et al. (1953a, (1.10.5), p. 25); with
Proof sketch:: 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 , 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 -plane.
Referenced by:: §25.11(i), §25.11(vii), Erratum (V1.2.1) for Figure 5.9.1
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 (see Figure 5.9.1) is a loop around the negative real axis as described for (25.5.20).

§25.11(viii) Further Integral Representations

ⓘ

Notes:: See Adamchik (1998).
Referenced by:: §25.11(vi)
Permalink:: http://dlmf.nist.gov/25.11.viii
Addition (effective with 1.1.4):: A line of text was added just above (25.11.33) reading “where $H_{n}$ are the harmonic numbers:”.
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.11 and Ch.25

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 , $\mathrm{e}$ : base of natural logarithm, $\cosh\NVar{z}$ : hyperbolic cosine function, $\int$ : integral, $\Re$ : real part, : real variable, : real or complex parameter and : complex variable
Keywords:: improper integral, integral representation
Proof sketch:: The improper integral can be evaluated by using $2\cosh x={\mathrm{e}}^{x}(1+{\mathrm{e}}^{-2x})$ , changing variables , and then applying (25.11.35).
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 $\int_{0}^{a}x^{n}\psi\left(x\right)\,\mathrm{d}x=(-1)^{n-1}\zeta'\left(-n% \right)+(-1)^{n}H_{n}\frac{B_{n+1}}{n+1}-\sum_{k=0}^{n}(-1)^{k}\genfrac{(}{)}{% 0.0pt}{}{n}{k}H_{k}\frac{B_{k+1}(a)}{k+1}a^{n-k}+\sum_{k=0}^{n}(-1)^{k}% \genfrac{(}{)}{0.0pt}{}{n}{k}\zeta'\left(-k,a\right)a^{n-k},$ $n=1,2,\dots$ , $\Re a>0$ ,

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\,\mathrm{d}\NVar{x}$ : differential of , $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, $H_{\NVar{n}}$ : harmonic number, $\int$ : integral, $\Re$ : real part, : nonnegative integer, : nonnegative integer, : real variable, : real or complex parameter and : harmonic number
Keywords:: definite integral, integral representation
Source:: Adamchik (1998, (17), p. 197)
Referenced by:: Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.11.E32
Encodings:: TeX, pMML, png
Notation (effective with 1.1.4):: The notation previously used for the harmonic number (resp. ) has been replaced to be $H_{n}$ (resp. $H_{k}$ ).
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.11(viii), §25.11 and Ch.25

where $H_{n}$ are the harmonic numbers:

25.11.33 $H_{n}=\sum_{k=1}^{n}k^{-1}.$

ⓘ

Defines:: $H_{\NVar{n}}$ : harmonic number
Symbols:: : nonnegative integer, : nonnegative integer and : harmonic number
Keywords:: harmonic number
Source:: Knuth (1968, Section 1.2.7, p. 73)
Referenced by:: §25.11(viii), §25.16(ii), Erratum (V1.1.4) for Notation
Permalink:: http://dlmf.nist.gov/25.11.E33
Encodings:: TeX, pMML, png
Notation (effective with 1.1.4):: The notation previously used for the harmonic number has been replaced to be $H_{n}$ .
Suggested 2021-08-23 by Gergő Nemes
See also:: Annotations for §25.11(viii), §25.11 and Ch.25

25.11.34 $n\int_{0}^{a}\zeta'\left(1-n,x\right)\,\mathrm{d}x=\zeta'\left(-n,a\right)-% \zeta'\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a\right)}{n(n+1)},$ $n=1,2,\dots$ , $\Re a>0$ .

ⓘ

Symbols:: $B_{\NVar{n}}$ : Bernoulli numbers, $B_{\NVar{n}}\left(\NVar{x}\right)$ : Bernoulli polynomials, $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\zeta\left(\NVar{s}\right)$ : Riemann zeta function, $\,\mathrm{d}\NVar{x}$ : differential of , $\int$ : integral, $\Re$ : real part, : nonnegative integer, : real variable and : real or complex parameter
Keywords:: definite integral, integral representation
Source:: Adamchik (1998, (15), p. 196)
Permalink:: http://dlmf.nist.gov/25.11.E34
Encodings:: TeX, pMML, png
See also:: Annotations for §25.11(viii), §25.11 and Ch.25

§25.11(ix) Integrals

ⓘ

Keywords:: Hurwitz zeta function, integrals
Permalink:: http://dlmf.nist.gov/25.11.ix
See also:: Annotations for §25.11 and Ch.25

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

ⓘ

Keywords:: Hurwitz zeta function, series representations
Notes:: See Apostol (1976, p. 249). For (25.11.35) use (25.11.25) and (25.11.8).
Referenced by:: Erratum (V1.1.3) for References
Permalink:: http://dlmf.nist.gov/25.11.x
Removal (effective with 1.1.4):: The line immediately below (25.11.36) which previously read “where $\chi(n)$ is a Dirichlet character $\pmod{k}$ (§27.8). Compare (25.15.1) and (25.15.3).” has been removed because (25.11.36) has been removed.
Addition (effective with 1.1.4):: Included mention of (8.15.2) in the last sentence of this subsection.
Addition (effective with 1.0.23):: A sentence just below (25.11.36) was added indicating that one should make a comparison with (25.15.1) and (25.15.3).
See also:: Annotations for §25.11 and Ch.25

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 , $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $\int$ : integral, $\Re$ : real part, : nonnegative integer, : real variable, : real or complex parameter and : complex variable
Keywords:: improper integral, infinite series, integral representation, series representation
Proof sketch:: 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 , 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 , (25.11.35) reduces to (25.2.3).

25.11.36 Removed because it is just (25.15.1) combined with (25.15.3).

ⓘ

Symbols:: $L\left(\NVar{s},\NVar{\chi}\right)$ : Dirichlet -function and : nonnegative integer
Referenced by:: §25.11(x), Erratum (V1.0.23) for Equation (25.11.36), Erratum (V1.0.23) for Equation (25.11.36)
Permalink:: http://dlmf.nist.gov/25.11.E36
Removal (effective with 1.1.4):: This equation has been removed because it is just (25.15.1) combined with (25.15.3).
Clarification (effective with 1.0.23):: The Dirichlet -function was inserted on the left-hand side. The upper-index of the finite sum which originally was , was replaced with since $\chi(k)=0$ .
See also:: Annotations for §25.11(x), §25.11 and Ch.25

See also (8.15.2) and Srivastava and Choi (2001).

§25.11(xi) Sums

ⓘ

Keywords:: Catalan’s constant, Hurwitz zeta function, Riemann zeta function, sums
Notes:: See Adamchik and Srivastava (1998).
Permalink:: http://dlmf.nist.gov/25.11.xi
See also:: Annotations for §25.11 and Ch.25

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, : nonnegative integer, : nonnegative integer and : 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 $\sum_{k=1}^{\infty}\genfrac{(}{)}{0.0pt}{}{n+k}{k}\zeta\left(n+k+1,a\right)z^{% k}=\frac{(-1)^{n}}{n!}\left({\psi}^{(n)}\left(a\right)-{\psi}^{(n)}\left(a-z% \right)\right),$ $n=1,2,3,\dots$ , $\Re a>0$ ,

ⓘ

Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\genfrac{(}{)}{0.0pt}{}{\NVar{m}}{\NVar{n}}$ : binomial coefficient, $\psi\left(\NVar{z}\right)$ : psi (or digamma) function, : factorial (as in ), $\Re$ : real part, : nonnegative integer, : nonnegative integer, : real or complex parameter and : complex variable
Keywords:: infinite series, series representation
Source:: Adamchik and Srivastava (1998, (2.29), p. 138)
Permalink:: http://dlmf.nist.gov/25.11.E38
Encodings:: TeX, pMML, png
See also:: Annotations for §25.11(xi), §25.11 and Ch.25

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, : nonnegative integer and : 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 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, : nonnegative integer and : 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) -Asymptotic Behavior

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Keywords:: Hurwitz zeta function, asymptotic expansions for large parameter, derivatives
Notes:: See Apostol (1952) and Elizalde (1986).
Referenced by:: §25.11(vi), Erratum (V1.2.1) for §25.11(xii)
Permalink:: http://dlmf.nist.gov/25.11.xii
Correction and Addition (effective with 1.2.1):: Just above (25.11.44), previously it was mentioned that $a\to\infty$ is in the sector “ $|\operatorname{ph}a|\leq\frac{1}{2}\pi-\delta(<\frac{1}{2}\pi)$ ”. This has been updated to be “ $|\operatorname{ph}a|\leq\pi-\delta(<\pi)$ (see Nemes (2017a))”. Also, just below (25.11.45), it was mentioned that “For error bounds for (25.11.43), (25.11.44) and (25.11.45), see Nemes (2017a).”
See also:: Annotations for §25.11 and Ch.25

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

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

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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, : real or complex parameter and : complex variable
Keywords:: asymptotic approximation
Source:: Apostol (1952, (15), p. 6)
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 fixed, $\Re s>1$ ,

25.11.42 $\zeta\left(s,\alpha+i\beta\right)\to 0,$

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Symbols:: $\zeta\left(\NVar{s},\NVar{a}\right)$ : Hurwitz zeta function, $\mathrm{i}$ : imaginary unit and : complex variable
Keywords:: asymptotic approximation
Source:: Apostol (1952, (16), p. 6)
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 $\zeta\left(s,a\right)-\frac{a^{1-s}}{s-1}-\frac{1}{2}a^{-s}\sim\sum_{k=1}^{% \infty}\frac{B_{2k}}{(2k)!}{\left(s\right)_{2k-1}}a^{1-s-2k}.$

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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 ), : nonnegative integer, : real or complex parameter and : complex variable
Keywords:: asymptotic approximation
Source:: Paris (2005b, (1.3), (1.4), p. 298)
Referenced by:: §25.11(xii), §25.11(xii), §25.11(xii), Erratum (V1.0.9) for Chapters 7, 25, Erratum (V1.2.1) for Addition
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 $|\operatorname{ph}a|\leq\pi-\delta(<\pi)$ .

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, : nonnegative integer and : real or complex parameter
Keywords:: asymptotic approximation
Sources:: Elizalde (1986, (18), p. 349); Nemes (2017a, for the region of validity)
Referenced by:: §25.11(xii), §25.11(xii), Erratum (V1.2.1) for §25.11(xii), Erratum (V1.2.1) for Addition
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 $\zeta'\left(-2,a\right)-\frac{1}{12}a+\frac{1}{9}a^{3}-\left(\frac{1}{6}a-% \frac{1}{2}a^{2}+\frac{1}{3}a^{3}\right)\ln a\sim\sum_{k=1}^{\infty}\frac{2B_{% 2k+2}}{(2k+2)(2k+1)2k(2k-1)}a^{-(2k-1)}.$

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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, : nonnegative integer and : real or complex parameter
Keywords:: asymptotic approximation
Sources:: Elizalde (1986, (19), p. 350); Nemes (2017a, for the region of validity)
Referenced by:: §25.11(xii), §25.11(xii), Erratum (V1.2.1) for Addition
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 error bounds for (25.11.43), (25.11.44) and (25.11.45), see Nemes (2017a).