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Hurwitz zeta function

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11: Bibliography C
  • M. W. Coffey (2008) On some series representations of the Hurwitz zeta function. J. Comput. Appl. Math. 216 (1), pp. 297–305.
  • M. W. Coffey (2009) An efficient algorithm for the Hurwitz zeta and related functions. J. Comput. Appl. Math. 225 (2), pp. 338–346.
  • 12: Bibliography B
  • B. C. Berndt (1972) On the Hurwitz zeta-function. Rocky Mountain J. Math. 2 (1), pp. 151–157.
  • 13: Bibliography M
  • J. Miller and V. S. Adamchik (1998) Derivatives of the Hurwitz zeta function for rational arguments. J. Comput. Appl. Math. 100 (2), pp. 201–206.
  • 14: Software Index
    Open Source With Book Commercial
    25.21(iv) ζ ( s , a ) a
    ‘✓’ indicates that a software package implements the functions in a section; ‘a’ indicates available functionality through optional or add-on packages; an empty space indicates no known support. … In the list below we identify four main sources of software for computing special functions. …
  • Commercial Software.

    Such software ranges from a collection of reusable software parts (e.g., a library) to fully functional interactive computing environments with an associated computing language. Such software is usually professionally developed, tested, and maintained to high standards. It is available for purchase, often with accompanying updates and consulting support.

  • The following are web-based software repositories with significant holdings in the area of special functions. …
    15: Errata
  • Equations (25.11.6), (25.11.19), and (25.11.20)


    Originally all six integrands in these equations were incorrect because their numerators contained the function B ~ 2 ( x ) . The correct function is B ~ 2 ( x ) - B 2 2 . The new equations are:

    25.11.6 ζ ( s , a ) = 1 a s ( 1 2 + a s - 1 ) - s ( s + 1 ) 2 0 B ~ 2 ( x ) - B 2 ( x + a ) s + 2 d x , s 1 , s > - 1 , a > 0

    Reported 2016-05-08 by Clemens Heuberger.

    25.11.19 ζ ( s , a ) = - ln a a s ( 1 2 + a s - 1 ) - a 1 - s ( s - 1 ) 2 + s ( s + 1 ) 2 0 ( B ~ 2 ( x ) - B 2 ) ln ( x + a ) ( x + a ) s + 2 d x - ( 2 s + 1 ) 2 0 B ~ 2 ( x ) - B 2 ( x + a ) s + 2 d x , s > - 1 , s 1 , a > 0

    Reported 2016-06-27 by Gergő Nemes.

    25.11.20 ( - 1 ) k ζ ( k ) ( s , a ) = ( ln a ) k a s ( 1 2 + a s - 1 ) + k ! a 1 - s r = 0 k - 1 ( ln a ) r r ! ( s - 1 ) k - r + 1 - s ( s + 1 ) 2 0 ( B ~ 2 ( x ) - B 2 ) ( ln ( x + a ) ) k ( x + a ) s + 2 d x + k ( 2 s + 1 ) 2 0 ( B ~ 2 ( x ) - B 2 ) ( ln ( x + a ) ) k - 1 ( x + a ) s + 2 d x - k ( k - 1 ) 2 0 ( B ~ 2 ( x ) - B 2 ) ( ln ( x + a ) ) k - 2 ( x + a ) s + 2 d x , s > - 1 , s 1 , a > 0

    Reported 2016-06-27 by Gergő Nemes.

  • 16: Bibliography K
  • A. A. Karatsuba and S. M. Voronin (1992) The Riemann Zeta-Function. de Gruyter Expositions in Mathematics, Vol. 5, Walter de Gruyter & Co., Berlin.
  • J. Keating (1993) The Riemann Zeta-Function and Quantum Chaology. In Quantum Chaos (Varenna, 1991), Proc. Internat. School of Phys. Enrico Fermi, CXIX, pp. 145–185.
  • M. K. Kerimov (1980) Methods of computing the Riemann zeta-function and some generalizations of it. USSR Comput. Math. and Math. Phys. 20 (6), pp. 212–230.
  • K. S. Kölbig (1970) Complex zeros of an incomplete Riemann zeta function and of the incomplete gamma function. Math. Comp. 24 (111), pp. 679–696.
  • K. S. Kölbig (1972a) Complex zeros of two incomplete Riemann zeta functions. Math. Comp. 26 (118), pp. 551–565.