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Computational strategies for the Riemann zeta function. (English) Zbl 0972.11077
This fifty page paper provides a compendium of evaluation methods for the Riemann zeta-function. Formulas are given that range from historical attempts to recent found convergent series, with some curious oddities, old and new ones. The first section is devoted to provide a motivation for the search of efficient evaluation schemes. In the second, some properties of the Riemann zeta-function are listed, and the third is devoted to evaluations for general complex arguments. In the fourth section, rational zeta series are considered, while the fifth section focusses on integer arguments, specially on positive odd integer arguments, since the calculations for even ones can proceed through existing fast algorithms for computation of $\pi$ and its powers. Section six deals with alternative value-recycling schemes that can be successfully invoked for certain sets of arguments with integer differences. Section seven is devoted to evaluations of $\zeta$-values for integer arguments and in certain arithmetic progressions, with some detailed comments on the complexity issue. Finally, section eight contains some curiosities and open questions. The paper, very valuable as a reference, concentrates primarily on practical computational issues, such issues depending on the domain of the argument, the speed of computation one wishes to achieve, and the incidence of what the author calls in the paper “value recycling”. It should be mentioned that, in some way, in the reviewer’s work: Ten physical applications of spectral zeta-functions. Lect. Notes Phys., New Ser. m35, Springer, Berlin (1995; Zbl 0855.00002), Commun. Math. Phys. 198, 83-95 (1998; Zbl 0932.11056) and J. Comput. Appl. Math. 118, 125-142 (2000; Zbl 1016.11034), there have been implemented similar ideas to other zeta functions, as the Hurwitz and Epstein zeta-functions, and generalizations thereof.

MSC:
11M06$\zeta (s)$ and $L(s, \chi)$
11Y35Analytic computations
11-02Research monographs (number theory)
WorldCat.org
Full Text: DOI
References:
[1] Adamchik, V. S.; Srivastava, H. M.: Some series of the zeta and related functions. Analysis 18, 131-144 (1998) · Zbl 0919.11056
[2] Aizenberg, L.; Adamchik, V.; Levit, V.: Approaching the Riemann hypothesis with Mathematica. Math. J. 7, 54-57 (1997)
[3] Akiyama, S.; Tanigawa, Y.: Calculation of values of L-functions associated to elliptic curves. Math. comp. 68, 1201-1231 (1999) · Zbl 0923.11100
[4] G. Almkvist, A. Granville, Borwein and Bradley’s Apéry-like formulae for {$\zeta$}(4n+3), Exp. Math. 8 (1999) 197--203. · Zbl 0976.11035
[5] Amdeberhan, T.; Zeilberger, D.: Hypergeometric series acceleration via the WZ method. Electron. J. Combin. 4, No. 3, #R3 (1996) · Zbl 0884.05010
[6] Apostol, T.: Introduction to analytic number theory. (1976) · Zbl 0335.10001
[7] E. Bach, Comments on the complexity of some number-theoretic constants, manuscript, 28 July 1992.
[8] Bach, E.: The complexity of number-theoretic constants. Inform. process. Lett. 62, 145-152 (1997) · Zbl 1053.11544
[9] Bach, E.; Lukes, R.; Shallit, J.; Williams, H.: Results and estimates on pseudopowers. Math. comp. 65, 1737-1747 (1996) · Zbl 0853.11103
[10] Bach, E.; Shallit, J.: Algorithmic number theory, vol. I. (1996) · Zbl 0873.11070
[11] Bailey, D.; Borwein, J.; Crandall, R.: On the Khintchine constant. Math. comp. 66, 417-431 (1997) · Zbl 0854.11078
[12] Bailey, D.; Borwein, P.; Plouffe, S.: On the rapid computation of various polylogarithmic constants. Math. comp. 66, 903-913 (1997) · Zbl 0879.11073
[13] D. Bailey, R. Crandall, On the random character of fundamental constant expansions, manuscript, http://www.perfsci.com/free/techpapers. · Zbl 1047.11073
[14] Balazard, M.; Saias, E.; Yor, M.: Notes sur la fonction ${\zeta}$ de Riemann, 2. Adv. = 80 math 143, 284-287 (1999) · Zbl 0937.11032
[15] C. Bays, R. Hudson, Zeroes of Dirichlet L-functions and irregularities in the distribution of primes, Math. Comp. S 0025-5718(99)01105-9 (1999), elec. pub. 10 Mar.
[16] B.C. Berndt, Ramanujan’s Notebooks: Part II, Springer, Berlin, 1985, p. 359. · Zbl 0555.10001
[17] Berry, M. V.: Quantum chaology. Proc. roy. Soc. London A 413, 183-198 (1987)
[18] Berry, M. V.; Keating, J. P.: A new asymptotic representation for ${\zeta}$(1/2 + it) and quantum spectral determinants. Proc. roy. Soc. London A 437, 151-173 (1992) · Zbl 0776.11048
[19] Bombieri, E.: Remarks on the analytic complexity of zeta functions. London mathematical society lecture note series 247, 21-30 (1998)
[20] Bombieri, E.; Lagarias, J.: Complements to Li’s criterion for the Riemann hypothesis. J. number theory 77, 274-287 (1999) · Zbl 0972.11079
[21] Boo, R. C.: An elementary proof of \sumn$\geqslant 1$ 1/n2 = 3D ${\pi}2$/6. Amer. math. Mon. 94, 662-663 (1987) · Zbl 0624.40001
[22] Borwein, J. M.; Borwein, P. B.: Pi and the AGM. (1987)
[23] D. Borwein, P. Borwein, A. Jakinovski, An efficient algorithm for the Riemann Zeta function, available from the CECM preprint server, URL = 3D http://www.cecm.sfu.ca/preprints/1995pp.html, CECM-95-043.
[24] J. Borwein, D. Bradley, Searching symbolically for Apéry-like formulae for values of the Riemann Zeta function, SIGSAM Bull. Symbolic Algebraic Manipulation 116 (1996) 2--7. Also available from the CECM preprint server, URL = 3D http://www.cecm.sfu.ca/preprints/1996pp.html, CECM-96-076.
[25] J. Borwein, D. Bradley, Empirically determined Apéry-like formulae for {$\zeta$}(4n+3), Exp. Math. 6(3) (1997) 181--194. Also available from the CECM preprint server, URL = 3D http://www.cecm.sfu.ca/preprints /1996pp.html, CECM-96-069.
[26] Borwein, J. M.; Zucker, I. J.: Elliptic integral evaluation of the gamma function at rational values of small denominator. IMA J. Numer. anal. 12, 519-526 (1992) · Zbl 0758.65008
[27] P. Borwein, An efficient algorithm for the Riemann Zeta function, manuscript, 1995.
[28] L. de Branges, Proof of the Riemann hypothesis, 1999. ftp://ftp.math.purdue.edu/pub/branges/proof.ps
[29] Bredihin, B.: Applications of the dispersion method in binary additive problems. Dokl. akad. Nauk. SSSR 149, 9-11 (1963)
[30] R.P. Brent, The complexity of multi-precision arithmetic. Complexity of Computational Problem Solving, manuscript, 1976. · Zbl 0342.65031
[31] Brent, R. P.: Fast multiple-precision evaluation of elementary functions. J. ACM 23, 242-251 (1976) · Zbl 0324.65018
[32] Brent, R. P.: On the zeros of the Riemann zeta function in the critical strip. Math. comp. 33, 1361-1372 (1979) · Zbl 0422.10031
[33] Brent, R. P.; Van De Lune, J.; Te Riele, H. J. J.; Winter, D. T.: On the zeros of the Riemann zeta function in the critical strip. II. Math. comp. 39, 681-688 (1982) · Zbl 0486.10028
[34] Brent, R. P.; Mcmillan, E. M.: Some new algorithms for high-precision calculation of Euler’s constant. Math. comp. 34, 305-312 (1980) · Zbl 0442.10002
[35] D.J. Broadhurst, Polylogarithmic ladders, hypergeometric series and the ten millionth digits of {$\zeta$}(3) and {$\zeta$}(5), manuscript: OUT-4102-71, math.CA/9803067 (16 March 1998).
[36] Buhler, J.; Crandall, R.; Ernvall, R.; Metsänkylä, T.: Irregular primes to four million. Math. comp. 61, 151-153 (1993) · Zbl 0789.11020
[37] J. Buhler, R. Crandall, R. Ernvall, T. Metsankyla, A. Shokrollahi, Irregular primes and cyclotomic invariants to eight million, manuscript, 1996.
[38] Buhler, J.; Crandall, R.; Sompolski, R. W.: Irregular primes to one million. Math. comp. 59, 717-722 (1992) · Zbl 0768.11009
[39] D. Bump, K.K. Choi, P. Kurlberg, J. Vaaler, A local Riemann hypothesis, I, Math. Z. 233 (2000) 1--19.
[40] Cohen, H.; Olivier, M.: Calcul des valeurs de la fonction zêta de Riemann en multiprécision. CR acad. Sci. sér. I math. 314, No. 6, 427-430 (1992) · Zbl 0751.11057
[41] Comtet, L.: Advanced combinatorics. (1974) · Zbl 0283.05001
[42] A. Connes, Trace formula in noncommutative Geometry and the zeros of the Riemann zeta function, manuscript, http://xxx.lanl.gov/ps/math/9811068, 1988.
[43] Crandall, R. E.: Mathematica for the sciences. (1991)
[44] Crandall, R. E.: Pojects in scientific computation. (1994) · Zbl 0791.65001
[45] Crandall, R. E.: Topics in advanced scientific computation. (1996) · Zbl 0844.65001
[46] Crandall, R. E.: On the quantum zeta function. J. phys. A. math. Gen. 29, 6795-6816 (1996) · Zbl 0905.58040
[47] R.E. Crandall, Fast evaluation of Epstein zeta functions, manuscript, at http://www.perfsci.com, October 1998.
[48] R.E. Crandall, Recycled (simultaneous) evaluations of the Riemann zeta function, manuscript, 1999.
[49] R.E. Crandall, Alternatives to the Riemann--Siegel formula, manuscript, 1999.
[50] Crandall, R. E.; Buhler, J. P.: On the evaluation of Euler sums. Exp. math. 3, No. 4, 275-285 (1995) · Zbl 0833.11045
[51] R. Crandall, C. Pomerance, Prime Numbers: A Computational Perspective, Springer, Berlin, manuscript. · Zbl 1088.11001
[52] Dilcher, K.: Sums of products of Bernoulli numbers. J. number theory 60, No. 1, 23-41 (1996) · Zbl 0863.11011
[53] Dutt, A.; Rokhlin, V.: Fast Fourier transforms for nonequispaced data. SIAM J. Comput. 14, No. 6, 1368-1393 (1993) · Zbl 0791.65108
[54] Dutt, A.; Gu, M.; Rokhlin, V.: Fast algorithms for polynomial interpolation, integration, and differentiation. SIAM J. Numer. anal. 33, No. 5, 1689-1711 (1996) · Zbl 0862.65005
[55] Edwards, H. M.: Riemann’s zeta function. (1974) · Zbl 0315.10035
[56] P. Flajolet, I. Vardi, Zeta function expansions of classical constants, 1996 24 Feb, manuscript. http://pauillac.inria.fr/algo/flajolet/Publications/publist.html
[57] W. Gabcke, Neue Herleitung und Explizite Restabschätzung der Riemann--Siegel--Formel, Thesis, Georg-August-Universität zu Göttingen, 1979. · Zbl 0499.10040
[58] W. Galway, Fast computation of the Riemann Zeta function to abitrary accuracy, manuscript, 1999.
[59] Haible, B.; Papanikolaou, T.: Fast multiprecision evaluation of series of rational numbers. Lecture notes in computer science 1423 (1998) · Zbl 1067.11517
[60] Henrici, P.: Applied and computational complex analysis, vol. 2. (1977) · Zbl 0363.30001
[61] M.M. Hjortnaes, Overf{$\phi$}ring av rekken \sumk=3D1\infty (1/k3) till et bestemt integral, Proceedings of 12th Congress on Scand. Maths, Lund 1953, (Lund 1954).
[62] A.E. Ingham, The Distribution of Prime Numbers 2nd edition Cambridge Tracts in Mathematics, Vol. 30, Cambridge University Press, Cambridge 1932, 1990, with a Foreword by R.C. Vaughan. · Zbl 0006.39701
[63] Ivic, A.: The Riemann zeta-function. (1985) · Zbl 0455.10025
[64] Ivic, A.: On some results concerning the Riemann hypothesis. Lecture note series 247, 139-167 (1998)
[65] Karatsuba, E. A.: On new method of fast computation of transcendental functions. Uspechi mat. Nauk (Successes of mathematical sciences) 46, No. 2, 219-220 (1991)
[66] Karatsuba, E. A.: On fast computation of transcendental functions. Soviet math. Dokl. 43, 693-694 (1991) · Zbl 0773.65007
[67] Karatsuba, E. A.: Fast evaluation of transcendental functions. Probl. peredachi inform. 27, 76-99 (1991) · Zbl 0754.65021
[68] Karatsuba, E. A.: Fast evaluation of ${\zeta}$(3). Probl. peredachi inform. 29, 68-73 (1993) · Zbl 0791.11073
[69] Karatsuba, E. A.: Fast calculation of the Riemann zeta function ${\zeta}(s)$ for integer values of the argument s. Probl. inform. Transm. 31, 353-362 (1995) · Zbl 0895.11032
[70] Karatsuba, E. A.: On fast computation of Riemann zeta function for integer values of argument. Dokl.ran 349, No. 4, 463 (1996) · Zbl 0923.11172
[71] Karatsuba, E. A.: Fast evaluation of hypergeometric function by FEE. Series in approximation and decomposition, 303, 314 =20 (1999) · Zbl 1017.65014
[72] Karatsuba, A. A.; Voronin, S. M.: The Riemann zeta-function. (1992) · Zbl 0756.11022
[73] Koecher, M.: Klassische elementare analysis. (1987) · Zbl 0624.26001
[74] Koecher, M.: Lett. math. Intelligencer. 2, 62-64 (1980)
[75] Lagarias, J.: On a positivity property of the Riemann ${\xi}$-function. Acta arithmetica 89, 217-234 (1999) · Zbl 0928.11035
[76] Lagarias, J.; Odlyzko, A.: Computing ${\pi}(x)$: an analytic method. J. algorithms 8, 173-191 (1987) · Zbl 0622.10027
[77] Lan, Y.: A limit formula for ${\zeta}$(2k+1). J. number theory = 20 78, 271-286 (1999) · Zbl 0949.11041
[78] Landen, J.: Math. memoirs. 1, 118 (1780)
[79] Lewin, L.: Polylogarithms and associated functions. (1981) · Zbl 0465.33001
[80] Li, X. -J.: The positivity of a sequence of numbers and the Riemann hypothesis. J. number theory 65, 325-333 (1997) · Zbl 0884.11036
[81] Littlewood, J. E.: Sur la distribution des nombres premiers. CR acad. Sci. (Paris) 158, 1869-1872 (1914) · Zbl 45.0305.01
[82] Van De Lune, J.; Te Riele, H. J. J.; Winter, D. T.: On the zeros of the Riemann zeta function in the critical strip. IV. Math. comp. 46, 667-681 (1986) · Zbl 0585.10023
[83] Motohashi, Y.: Spectral theory of the Riemann zeta-function, Cambridge tracts in mathematics, vol. 127. (1997) · Zbl 0878.11001
[84] Odlyzko, A.: On the distribution of spacings between zeros of the zeta function. Math. comp. 48, 273-308 (1991) · Zbl 0615.10049
[85] Odlyzko, A.: Analytic computations in number theory. Proc. symp. Appl. math. 48, 441-463 (1994) · Zbl 0822.11085
[86] A. Odlyzko, The 1020th zero of the Riemann zeta function and 175 million of its neighbors, 1998 http://www.research.att.com/ amo/unpublished/index.html.
[87] Odlyzko, A.; Schönhage, A.: Fast algorithms for multiple evaluations of the Riemann zeta-function. Trans. amer. Math. soc. 309, 797-809 (1988) · Zbl 0706.11047
[88] Odlyzko, A.; Te Riele, H. J. J.: Disproof of the mertens conjecture. J. reine angew. Math. 357, 138-160 (1985) · Zbl 0544.10047
[89] Paris, R. B.: An asymptotic representation for the Riemann zeta function on the critical line. Proc. roy. Soc. London ser. A 446, 565-587 (1994) · Zbl 0827.11051
[90] S. Plouffe, http://www.lacim.uqam.ca/pi/index.html (1998).
[91] Pour-El, M. B.; Richards, I.: The wave equation and computable initial data such that its unique solution is not computable. Adv. math. 39, 215-239 (1981) · Zbl 0465.35054
[92] Pustyl’nikov, L. D.: On a property of the classical zeta-function associated with the Riemann conjecture on zeros. Russian math. Surveys 54, No. 1, 262-263 (1999)
[93] R. Ramanujan, in: G. Hardy, P. Aiyar, B. Wilson (Eds.), Collected Papers of Srinivasa Ramanujan, Cambridge University Press, Cambridge, 1927, Section 19. · Zbl 53.0030.02
[94] Ribenboim, P.: The new book of prime number records. (1995) · Zbl 0856.11001
[95] H. Riesel, Prime Numbers and Computer Methods for Factorization, Birkhauser, Boston, 1994, p. 126.
[96] M.O. Rubinstein, Evidence for a spectral interpretation of the zeros of L-functions, Dissertation, Princeton University, 1998.
[97] Schönhage, A.: Schnelle multiplikation grosser zahlen. Computing 7, 281-292 (1971)
[98] Skewes, S.: On the difference ${\pi}(x) - Li(x)$. J. London math. Soc. 8, 277-283 (1933) · Zbl 0007.34003
[99] Skewes, S.: On the difference ${\pi}(x) - Li(x)$, II. J. London math. Soc. 5, 48-70 (1955) · Zbl 0068.26802
[100] Te Riele, H. J. J.: On the sign of the difference of ${\pi}(x) - Li(x)$. Math. comp. 48, 323-328 (1987) · Zbl 0612.10035
[101] Titchmarsh, E. C.: The theory of the Riemann zeta-function. (1967) · Zbl 0042.07901
[102] Vardi, I.: Computational recreations in Mathematica. (1991) · Zbl 0786.11002
[103] Van Der Pol, B.: Bull. AMS. 53, 976 ff (1947)
[104] Van Der Poorten, A.: A proof that Euler missed: apéry’s proof of the irrationality of ${\zeta}$(3). Math. intelligencer 1, 195-203 (1970) · Zbl 0409.10028
[105] Voros, A.: Nucl. phys.. 165, 209-236 (1980)
[106] Whittaker, E. T.; Watson, G. N.: A course of modern analysis. (1969) · Zbl 45.0433.02
[107] S. Wolfram, D. Hillman, private communication, 1998.
[108] S.C. Woon, Generalization of a relation between the Riemann zeta function and the Bernoulli numbers, 1998 December, manuscript.
[109] Wrench, J. W.; Shanks, D.: Questions concerning Khintchine’s constant and the efficient computation of regular continued fractions. Math. comp. 20, 444-448 (1966) · Zbl 0222.65006
[110] Zhang, N. Y.; Williams, K. S.: Some series representations of ${\zeta}$(2n+1). Rocky mountain J. Math. 23, No. 4, 1581-1592 (1993) · Zbl 0808.11053