LEONHARD EULER (1707-1783) AND THE COMPUTATIONAL ASPECTS OF SOME ZETA-FUNCTION SERIES

- Journal title : Journal of the Korean Mathematical Society
- Volume 44, Issue 5, 2007, pp.1163-1184
- Publisher : The Korean Mathematical Society
- DOI : 10.4134/JKMS.2007.44.5.1163

Title & Authors

LEONHARD EULER (1707-1783) AND THE COMPUTATIONAL ASPECTS OF SOME ZETA-FUNCTION SERIES

Srivastava, Hari Mohan;

Srivastava, Hari Mohan;

Abstract

In this presentation dedicated to the tricentennial birth anniversary of the great eighteenth-century Swiss mathematician, Leonhard Euler (1707-1783), we begin by remarking about the so-called Basler problem of evaluating the Zeta function [in the much later notation of Georg Friedrich Bernhard Riemann (1826-1866)] when s=2, which was then of vital importance to Euler and to many other contemporary mathematicians including especially the Bernoulli brothers [Jakob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748)], and for which a fascinatingly large number of seemingly independent solutions have appeared in the mathematical literature ever since Euler first solved this problem in the year 1736. We then investigate various recent developments on the evaluations and representations of when being the set of natural numbers. We emphasize upon several interesting classes of rapidly convergent series representations for which have been developed in recent years. In two of many computationally useful special cases considered here, it is observed that can be represented by means of series which converge much more rapidly than that in Euler's celebrated formula as well as the series used recently by Roger (1916-1994) in his proof of the irrationality of . Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of one of these series are capable of producing an accuracy of seven decimal places.

Keywords

analytic number theory;Riemann zeta function;Hurwitz(orgeneralized) zeta function;series representations;harmonic numbers;Bernoulli numbers and polynomials;generating functions;Euler numbers and polynomials;inductive argument;symbolic and numerical computations;

Language

English

Cited by

References

1.

H. Alzer, D. Karayannakis, and H. M. Srivastava, Series representations for some mathematical constants, J. Math. Anal. Appl. 320 (2006), no. 1, 145-162

2.

R. Apery, Irrationaliie de $\zeta$ (2) et $\zeta$ (3), Asterisque 61 (1979), 11-13

4.

J. M. Borwein, D. M. Bradley, and R. E. Crandall, Computational strategies for the Riemann zeta function, J. Comput. Appl. Math. 121 (2000), no. 1-2, 247-296

5.

M.-P. Chen and H. M. Srivastava, Some families of series representations for the Riemann $\zeta$ (3), Results Math. 33 (1998), no. 3-4,179-197

6.

J. Choi, Y. J. Cho, and H. M. Srivastava, Series involving the zeta function and multiple gamma functions, Appl. Math. Comput. 159 (2004), no. 2, 509-537

7.

J. Choi and H. M. Srivastava, Certain families of series associated with the Hurwitz-Lerch zeta function, Appl. Math. Comput. 170 (2005), no. 1, 399-409

8.

J. Choi and H. M. Srivastava, Explicit evaluation of Euler and related sums, Ramanujan J. 10 (2005), no. 1, 51-70

9.

J. Choi, H. M. Srivastava, and V. S. Adamchik, Multiple gamma and related functions, Appl. Math. Comput. 134 (2003), no. 2-3, 515-533

10.

D. Cvijovic and J. Klinowski, New rapidly convergent series representations for $\zeta$ (2n + 1), Proc. Amer. Math. Soc. 125 (1997), no. 5, 1263-1271

11.

A. Dabrowski, A note on values of the Riemann zeta function at positive odd integers, Nieuw Arch. Wisk. (4) 14 (1996), no. 2, 199-207

12.

A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher transcendental functions. Vols. I, II, Based, in part, on notes left by Harry Bateman. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1953

13.

J. A. Ewell, A new series representation for $\zeta$ (3), Amer. Math. Monthly 97 (1990), no. 3, 219-220

14.

J. A. Ewell, On the zeta function values $\zeta$ (2k + 1), k = 1,2, ... , Rocky Mountain J. Math. 25 (1995), no. 3, 1003-1012

15.

M. Garg, K. Jain, and H. M. Srivastava, Some relationships between the generalized Apostol-Bernoulli polynomials and Huruntz-Lercli zeta functions, Integral Transforms Spec. Funet. 17 (2006), no. 11, 803-815

16.

17.

R. W. Gosper, Jr., A calculus of series rearrangements, Algorithms and complexity (Proc. Sympos., Carnegie-Mellon Univ., Pittsburgh, Pa., 1976), pp. 121-151. Academic Press, New York, 1976

18.

E. R. Hansen, A Table of Series and Products, Englewood Cliffs, NJ, Prentice-Hall, 1975

19.

M. M. Hjortnaes, Overforing av rekken$\sum_{k=1}^{\infty}\;(1/k^3)$ til et bestemt integral, Proceedings of the Twelfth Scandanavian Mathematical Congress (Lund; August 10-15, 1953), pp. 211-213, Scandanavian Mathematical Society, Lund, 1954

20.

S. Kanemitsu, H. Kumagai, and M. Yoshimoto, Sums involving the Hurwitz zeta function, Ramanujan J. 5 (2001), no. 1,5-19

21.

S. Kanemitsu, H. Kumagai, H. M. Srivastava, and M. Yoshimoto, Some integral and asymptotic formulas associated with the Hurwitz zeta function, Appl, Math. Comput. 154 (2004), no. 3, 641-664

22.

N. Koblitz, p-adic numbers, p-adic analysis, and zeta-functions, Graduate Texts in Mathematics, Vol. 58. Springer-Verlag, New York-Heidelberg, 1977

23.

S.-D. Lin, H. M. Srivastava, and P.-Y. Wang, Some expansion formulas for a class of generalized Huruntz-Lerch. zeta functions, Integral Transforms Spec. Funet. 17 (2006), no. 11, 817-827

24.

W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and theorems for the special functions of mathematical physics, Third enlarged edition. Die Grundlehren der mathematischen Wissenschaften, Band 52 Springer-Verlag New York, Inc., New York 1966

25.

C. Nash and D. O'Connor, Ray-Singer torsion, topological field theories and the Riemann Zeta function at s = 3, in Low-Dimensional Topology and Quantum Field Theory (Proceedings of a NATO Advanced Research Workshop held at the Isaac Newton Institute at Cambridge, U.K.; September 6-12, 1992) (H. Osborn, Editor), pp. 279-288, Plenum Press, New York and London, 1993

26.

C. Nash and D. J. O'Connor, Ray-Singer torsion, topological field theories and the Riemann zeta function at s = 3, Low-dimensional topology and quantum field theory (Cambridge, 1992), 279-288, NATO Adv. Sci. Inst. Ser. B Phys., 315, Plenum, New York,1993

28.

H. M. Srivastava, A unified presentation of certain classes of series of the Riemann zeta function, Riv. Mat. Univ. Parma (4) 14 (1988), 1-23

29.

H. M. Srivastava, Sums of certain series of the Riemann zeta function, J. Math. Anal. Appl. 134 (1988), no. 1, 129-140

30.

H. M. Srivastava, Certain families of rapidly convergent series representations for $\zeta$ (2n+l), Math. Sci. Res. Hot-Line 1 (6) (1997), 1-6

31.

H. M. Srivastava, Further series representations for $\zeta$ (2n + 1), Appl. Math. Comput. 97 (1998), 1-15

32.

H. M. Srivastava, Some rapidly converging series for $\zeta$ (2n + 1), Proc. Amer. Math. Soc. 127 (1999), no. 2, 385-396

33.

H. M. Srivastava, Some simple algorithms for the evaluations and representations of the Riemann zeta function at positive integer arguments, J. Math. Anal. Appl. 246 (2000), no. 2, 331-351

34.

H. M. Srivastava and J. Choi, Series associated with the zeta and related functions, Kluwer Academic Publishers, Dordrecht, 2001

35.

H. M. Srivastava, M. L. Glasser, and V. S. Adamchik, Some definite integrals associated with the Riemann zeta function, Z. Anal. Anwendungen 19 (2000), no. 3, 831-846

36.

H. M. Srivastava and H. Tsumura, A certain class of rapidly convergent series representations for $\zeta$ (2n + 1), J. Comput. Appl. Math. 118 (2000), no. 1-2, 323-335

37.

H. M. Srivastava and H. Tsumura, New rapidly convergent series representations for $\zeta$ (2n + 1), L(2n, X) and L(2n + 1, X), Math. Sci. Res. Hot-Line 4 (2000), no. 7, 17-24

38.

H. M. Srivastava and H. Tsumura , Inductive construction of rapidly convergent series representations for $\zeta$ (2n+ 1), Int. J. Comput. Math. 80 (2003), no. 9, 1161-1173

39.

E. C. Titchmarsh, The theory of the Riemann zeta-function, Second edition, The Clarendon Press, Oxford University Press, New York, 1986

40.

F. G. Tricomi, Sulla somma delle inverse delle terze e quinte potenze dei numeri naiurali, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 47 (1969) 16-18

41.

H. Tsumura, On evaluation of the Dirichlet series at positive integers by q-calculation, J. Number Theory 48 (1994), no. 3, 383-391

42.

E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge University Press, Cambridge, 1996

43.

J. R. Wilton, A proof of Burnside's formula for log $\Gamma$ (x + 1) and certain allied properties of Riemann's $\zeta$ -function, Messenger of Math. 52 (1922) 90-93