Numerical Algorithms

, Volume 69, Issue 2, pp 253–270 | Cite as

Rigorous high-precision computation of the Hurwitz zeta function and its derivatives

  • Fredrik Johansson
Original Paper


We study the use of the Euler-Maclaurin formula to numerically evaluate the Hurwitz zeta function ζ(s, a) for \(s, a \in \mathbb {C}\), along with an arbitrary number of derivatives with respect to s, to arbitrary precision with rigorous error bounds. Techniques that lead to a fast implementation are discussed. We present new record computations of Stieltjes constants, Keiper-Li coefficients and the first nontrivial zero of the Riemann zeta function, obtained using an open source implementation of the algorithms described in this paper.


Hurwitz zeta function Riemann zeta function Arbitrary-precision arithmetic Rigorous numerical evaluation Fast polynomial arithmetic Power series 

PAC Codes

65D20 68W30 33F05 11-04 11M06 11M35 


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  1. 1.
    Adell, J.A.: Estimates of generalized Stieltjes constants with a quasi-geometric rate of decay. Proc. R. Soc. A 468, 1356–1370 (2012)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Bailey, D.H., Borwein, J.M. In: B. Engquist, W. Schmid, P. W. Michor (eds.) : Experimental mathematics: recent developments and future outlook, pp 51–66. Springer (2000)Google Scholar
  3. 3.
    Bernstein, D.J.: Fast multiplication and its applications. Algorithmic Number Theory 44, 325–384 (2008)Google Scholar
  4. 4.
  5. 5.
    Bogolubsky, A.I., Skorokhodov, S.L.: Fast evaluation of the hypergeometric function p F p−1(a;b;z) at the singular point z=1 by means of the Hurwitz zeta function ζ(α,s). Program. Comput. Softw. 32(3), 145–153 (2006)CrossRefMATHGoogle Scholar
  6. 6.
    Bohman, J., Fröberg C-E.: The Stieltjes function – definition and properties. Math. Comput. 51(183), 281–289 (1988)MATHGoogle Scholar
  7. 7.
    Borwein, J.M., Bradley, D.M., Crandall, R.E.: Computational strategies for the Riemann zeta function. J. Comput. Appl. Math. 121, 247–296 (2000)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Borwein, P.: An efficient algorithm for the Riemann zeta function. Canadian Mathematical Society Conference Proceedings 27, 29–34 (2000)Google Scholar
  9. 9.
    Brent, R.P., Kung, H.T.: Fast algorithms for manipulating formal power series. J. ACM 25(4), 581–595 (1978)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Coffey, M.W.: An efficient algorithm for the Hurwitz zeta and related functions. J. Comput. Appl. Math. 225(2), 338–346 (2009)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Arias de Reyna, J.: Asymptotics of Keiper-Li coefficients. Functiones et Approximatio Commentarii Mathematici 45(1), 7–21 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    The GMP development team: GMP: The GNU multiple precision arithmetic library.
  13. 13.
    The MPIR development team: MPIR: Multiple Precision Integers and Rationals.
  14. 14.
    Edwards H.M.: Riemann’s zeta function. Academic Press (1974)Google Scholar
  15. 15.
    Finck, T., Heinig, G., Rost, K.: An inversion formula and fast algorithms for Cauchy-Vandermonde matrices. Linear Algebra Appl. 183, 179–191 (1993)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Flajolet, P., Vardi, I.: Zeta function expansions of classical constants. Unpublished manuscript (1996).
  17. 17.
    Fousse, L., Hanrot, G., Lefèvre, V., Pélissier, P., Zimmermann, P.: MPFR: A Multiple-Precision Binary Floating-Point Library with Correct Rounding. ACM Trans. Math. Softw.” 33(2), 13:1–13:15 (June 2007) Google Scholar
  18. 18.
    Gould, H.: Series transformations for finding recurrences for sequences. Fibonacci Quarterly 28, 166–171 (1990)MATHMathSciNetGoogle Scholar
  19. 19.
    Haible, B., Papanikolaou, T.: Algorithmic Number Theory: Third International Symposium. In: Buhler, J. P. (ed.) Fast multiprecision evaluation of series of rational numbers, Vol. 1423, pp 338–350. Springer (1998)Google Scholar
  20. 20.
    Hart, W.B.: Fast Library for Number Theory: An Introduction, In: Proceedings of the Third international congress conference on Mathematical software, ICMS’10, pp 88–91. Springer-Verlag, Berlin, Heidelberg (2010). Google Scholar
  21. 21.
    Harvey, D., Brent, R.P.: Fast computation of Bernoulli, Tangent and Secant numbers. Springer Proceedings in Mathematics & Statistics 50, 127–142 (2013) MathSciNetGoogle Scholar
  22. 22.
    Hiary, G.: Fast methods to compute the Riemann zeta function. Ann. math. 174, 891–946 (2011)CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Wofram Research Inc: Some Notes on Internal Implementation (section of the online documentation for Mathematica 9.0) (2013).
  24. 24.
    Johansson, F.: Arb: a C library for ball arithmetic. ACM Communications in Computer Algebra 47, 166–169 (2013). DecemberCrossRefMATHGoogle Scholar
  25. 25.
    Keiper, J.B.: Power series expansions of Riemann’s ξ function. Math. Comput. 58 (198), 765–773 (1992)MATHMathSciNetGoogle Scholar
  26. 26.
    Knessl, C., Coffey, M.: An asymptotic form for the Stieltjes constants γ k(a) and for a sum S γ(n) appearing under the Li criterion. Math. Comput. 80(276), 2197–2217 (2011)CrossRefMATHMathSciNetGoogle Scholar
  27. 27.
    Knessl, C., Coffey, M.: An effective asymptotic formula for the Stieltjes constants. Math. Comput. 80(273), 379–386 (2011)CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Kreminski, R.: Newton-Cotes integration for approximating Stieltjes (generalized Euler) constants. Math. Comput. 72(243), 1379–1397 (2003)CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Li, X.-J.: The positivity of a sequence of numbers and the Riemann Hypothesis. Math. Comput. 65(2), 325–333 (1997)MATHGoogle Scholar
  30. 30.
    Matiyasevich, Y.: An artless method for calculating approximate values of zeros of Riemann’s zeta function (2012).
  31. 31.
    Matiyasevich, Y., Beliakov, G.: Zeroes of Riemann’s zeta function on the critical line with 20000 decimal digits accuracy. (2011)
  32. 32.
    Odlyzko, A.M., Schönhage, A.: Fast algorithms for multiple evaluations of the Riemann zeta function. Trans. the Am. Math. Soc. 309(2), 797–809 (1988)CrossRefMATHGoogle Scholar
  33. 33.
    Olver, F.W.J.: Asymptotics and Special Functions. A K Peters, Wellesley, MA (1997)MATHGoogle Scholar
  34. 34.
    Pétermann, Y.-F.S, Rémy, J.-L.: Arbitrary precision error analysis for computing ζ(s) with the Cohen-Olivier algorithm: complete description of the real case and preliminary report on the general case. Rapport de recherche, RR-5852, INRIA, 2006.
  35. 35.
    Stein, W.A.: Sage Mathematics Software. The Sage Development Team (2013).
  36. 36.
    van der Hoeven, J.: Making fast multiplication of polynomial numerically stable. Technical Report 2008-02, Université Paris-Sud, Orsay, France (2008)Google Scholar
  37. 37.
    van der Hoeven, J.: Ball arithmetic. Technical report, HAL, 2009.
  38. 38.
    Vepštas, L.: An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions. Numerical Algorithms 47(3), 211–252 (2008)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.RISCJohannes Kepler UniversityLinzAustria

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