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Hurwitz zeta function

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1: 25.11 Hurwitz Zeta Function
§25.11 Hurwitz Zeta Function
§25.11(i) Definition
§25.11(ii) Graphics
See accompanying text
Figure 25.11.1: Hurwitz zeta function ζ ( x , a ) , a = 0. … Magnify
See accompanying text
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 ζ ( s , a ) , the dilogarithm Li 2 ( z ) , the polylogarithm Li s ( z ) (also known as Jonquière’s function ϕ ( z , s ) ), Lerch’s transcendent Φ ( z , s , a ) , and the Dirichlet L -functions L ( s , χ ) .
3: 25.13 Periodic Zeta Function
Also,
25.13.2 F ( x , s ) = Γ ( 1 - s ) ( 2 π ) 1 - s ( e π i ( 1 - s ) / 2 ζ ( 1 - s , x ) + e π i ( s - 1 ) / 2 ζ ( 1 - s , 1 - x ) ) , 0 < x < 1 , s > 1 ,
25.13.3 ζ ( 1 - s , x ) = Γ ( s ) ( 2 π ) s ( e - π i s / 2 F ( x , s ) + e π i s / 2 F ( - x , s ) ) , 0 < x < 1 , s > 0 .
4: 25.18 Methods of Computation
§25.18(i) Function Values and Derivatives
Calculations relating to derivatives of ζ ( s ) and/or ζ ( s , a ) can be found in Apostol (1985a), Choudhury (1995), Miller and Adamchik (1998), and Yeremin et al. (1988). For the Hurwitz zeta function ζ ( s , a ) see Spanier and Oldham (1987, p. 653) and Coffey (2009). …
5: 25.19 Tables
  • Cloutman (1989) tabulates Γ ( s + 1 ) F s ( x ) , where F s ( x ) is the Fermi–Dirac integral (25.12.14), for s = - 1 2 , 1 2 , 3 2 , 5 2 , x = - 5 ( .05 ) 25 , to 12S.

  • Fletcher et al. (1962, §22.1) lists many sources for earlier tables of ζ ( s ) 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 ζ ( s , a ) , 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 ζ ( s , a ) 25.11) and the polylogarithm Li s ( z ) 25.12(ii)) are special cases:
    25.14.2 ζ ( s , a ) = Φ ( 1 , s , a ) , s > 1 , a 0 , - 1 , - 2 , ,
    7: 25.21 Software
    §25.21(iv) Hurwitz Zeta Function
    8: 25.15 Dirichlet L -functions
    25.15.3 L ( s , χ ) = k - s r = 1 k - 1 χ ( r ) ζ ( s , r k ) ,
    9: 25.12 Polylogarithms
    and
    25.12.13 Li s ( e 2 π i a ) + e π i s Li s ( e - 2 π i a ) = ( 2 π ) s e π i s / 2 Γ ( s ) ζ ( 1 - s , a ) ,
    10: Bibliography P
  • R. B. Paris (2005b) The Stokes phenomenon associated with the Hurwitz zeta function ζ ( s , a ) . Proc. Roy. Soc. London Ser. A 461, pp. 297–304.