# §25.11 Hurwitz Zeta Function

## §25.11(i) Definition

The function was introduced in Hurwitz (1882) and defined by the series expansion

25.11.1, .

has a meromorphic continuation in the -plane, its only singularity in being a simple pole at with residue 1. As a function of , with () fixed, is analytic in the half-plane . The Riemann zeta function is a special case:

For most purposes it suffices to restrict because of the following straightforward consequences of (25.11.1):

Most references treat real with .

## §25.11(ii) Graphics

 Figure 25.11.1: Hurwitz zeta function , = 0.3, 0.5, 0.8, 1, . The curves are almost indistinguishable for , approximately. Symbols: : Hurwitz zeta function, : real variable and : real or complex parameter Permalink: http://dlmf.nist.gov/25.11.F1 Encodings: pdf, png Figure 25.11.2: Hurwitz zeta function , , . Symbols: : Hurwitz zeta function, : real variable and : real or complex parameter Permalink: http://dlmf.nist.gov/25.11.F2 Encodings: VRML, X3D, pdf, png

## §25.11(iv) Series Representations

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

## §25.11(vi) Derivatives

### ¶ -Derivatives

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

## §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

When , (25.11.35) reduces to (25.2.3).

where is a Dirichlet character 27.8).

See also Srivastava and Choi (2001).

## §25.11(xi) Sums

25.11.37, .

where is Catalan’s constant:

25.11.40

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

## §25.11(xii) -Asymptotic Behavior

As with fixed, ,

25.11.42

uniformly with respect to bounded nonnegative values of .

As in the sector , with and fixed, we have the asymptotic expansion

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

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