# Hurwitz zeta function

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##### 1: 25.11 Hurwitz Zeta Function
###### §25.11(ii) Graphics Figure 25.11.1: Hurwitz zeta function ζ ⁡ ( x , a ) , a = 0. … Magnify Figure 25.11.2: Hurwitz zeta function ζ ⁡ ( x , a ) , - 19.5 ≤ x ≤ 10 , 0.02 ≤ a ≤ 1 . Magnify 3D Help
##### 2: 25.1 Special Notation
The main related functions are the Hurwitz zeta function $\zeta\left(s,a\right)$, the dilogarithm $\mathrm{Li}_{2}\left(z\right)$, the polylogarithm $\mathrm{Li}_{s}\left(z\right)$ (also known as Jonquière’s function $\phi\left(z,s\right)$), Lerch’s transcendent $\Phi\left(z,s,a\right)$, and the Dirichlet $L$-functions $L\left(s,\chi\right)$.
##### 3: 25.13 Periodic Zeta Function
Also,
25.13.2 $F\left(x,s\right)=\frac{\Gamma\left(1-s\right)}{(2\pi)^{1-s}}\*\left(e^{\pi i(% 1-s)/2}\zeta\left(1-s,x\right)+e^{\pi i(s-1)/2}\zeta\left(1-s,1-x\right)\right),$ $0, $\Re s>1$,
25.13.3 $\zeta\left(1-s,x\right)=\frac{\Gamma\left(s\right)}{(2\pi)^{s}}\left(e^{-\pi is% /2}F\left(x,s\right)+e^{\pi is/2}F\left(-x,s\right)\right),$ $0, $\Re s>0$.
##### 4: 25.18 Methods of Computation
###### §25.18(i) Function Values and Derivatives
Calculations relating to derivatives of $\zeta\left(s\right)$ and/or $\zeta\left(s,a\right)$ can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988). For the Hurwitz zeta function $\zeta\left(s,a\right)$ see Spanier and Oldham (1987, p. 653) and Coffey (2009). …
##### 5: 25.19 Tables
• Cloutman (1989) tabulates $\Gamma\left(s+1\right)F_{s}(x)$, where $F_{s}(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

• Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta\left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta\left(s,a\right)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

• ##### 6: 25.14 Lerch’s Transcendent
The Hurwitz zeta function $\zeta\left(s,a\right)$25.11) and the polylogarithm $\mathrm{Li}_{s}\left(z\right)$25.12(ii)) are special cases:
25.14.2 $\zeta\left(s,a\right)=\Phi\left(1,s,a\right),$ $\Re s>1$, $a\neq 0,-1,-2,\dots$,
##### 10: Bibliography P
• R. B. Paris (2005b) The Stokes phenomenon associated with the Hurwitz zeta function $\zeta(s,a)$ . Proc. Roy. Soc. London Ser. A 461, pp. 297–304.