Some rapidly converging series for 𝜁(2𝑛+1)

Srivastava, H.

doi:10.1090/S0002-9939-99-04945-X

Some rapidly converging series for $\zeta (2n+1)$
HTML articles powered by AMS MathViewer

by H. M. Srivastava PDF

Proc. Amer. Math. Soc. 127 (1999), 385-396 Request permission

Abstract:

For a natural number $n$, the author derives several families of series representations for the Riemann Zeta function $\zeta (2n+1)$. Each of these series representing $\zeta (2n+1)$ converges remarkably rapidly with its general term having the order estimate: \begin{equation*}O(k^{-2n-1}\cdot m^{-2k})\qquad (k\to \infty ;\quad m=2,3,4,6).\end{equation*} Relevant connections of the results presented here with many other known series representations for $\zeta (2n+1)$ are also pointed out.

References

Sam Perlis, Maximal orders in rational cyclic algebras of composite degree, Trans. Amer. Math. Soc. 46 (1939), 82–96. MR 15, DOI 10.1090/S0002-9947-1939-0000015-X
Raymond Ayoub, Euler and the zeta function, Amer. Math. Monthly 81 (1974), 1067–1086. MR 360116, DOI 10.2307/2319041
Bruce C. Berndt, Modular transformations and generalizations of several formulae of Ramanujan, Rocky Mountain J. Math. 7 (1977), no. 1, 147–189. MR 429703, DOI 10.1216/RMJ-1977-7-1-147
Bruce C. Berndt, Ramanujan’s notebooks. Part II, Springer-Verlag, New York, 1989. MR 970033, DOI 10.1007/978-1-4612-4530-8
P. L. Butzer and M. Hauss, Riemann zeta function: rapidly converging series and integral representations, Appl. Math. Lett. 5 (1992), no. 2, 83–88. MR 1154617, DOI 10.1016/0893-9659(92)90118-S
P. L. Butzer, C. Markett, and M. Schmidt, Stirling numbers, central factorial numbers, and representations of the Riemann zeta function, Results Math. 19 (1991), no. 3-4, 257–274. MR 1100673, DOI 10.1007/BF03323285
Henri Cohen, Généralisation d’une construction de R. Apéry, Bull. Soc. Math. France 109 (1981), no. 3, 269–281 (French). MR 680278
Djurdje Cvijović and Jacek Klinowski, New rapidly convergent series representations for $\zeta (2n+1)$, Proc. Amer. Math. Soc. 125 (1997), no. 5, 1263–1271. MR 1376755, DOI 10.1090/S0002-9939-97-03795-7
Andrzej Dąbrowski, A note on values of the Riemann zeta function at positive odd integers, Nieuw Arch. Wisk. (4) 14 (1996), no. 2, 199–207. MR 1402840
John A. Ewell, A new series representation for $\zeta (3)$, Amer. Math. Monthly 97 (1990), no. 3, 219–220. MR 1048434, DOI 10.2307/2324688
John A. Ewell, On values of the Riemann zeta function at integral arguments, Canad. Math. Bull. 34 (1991), no. 1, 60–66. MR 1108930, DOI 10.4153/CMB-1991-010-2
John A. Ewell, On the zeta function values $\zeta (2k+1)$, $k=1,2,\cdots$, Rocky Mountain J. Math. 25 (1995), no. 3, 1003–1012. MR 1357106, DOI 10.1216/rmjm/1181072201
J.W.L. Glaisher, Summations of certain numerical series, Messenger Math. 42 (1912/1913), 19–34.
Emil Grosswald, Die Werte der Riemannschen Zetafunktion an ungeraden Argumentstellen, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1970 (1970), 9–13 (German). MR 272725
Emil Grosswald, Comments on some formulae of Ramanujan, Acta Arith. 21 (1972), 25–34. MR 302600, DOI 10.4064/aa-21-1-25-34
E.R. Hansen, A Table of Series and Products, Prentice-Hall, Englewood Cliffs, New Jersey, 1975.
Dieter Klusch, On the Taylor expansion of the Lerch zeta-function, J. Math. Anal. Appl. 170 (1992), no. 2, 513–523. MR 1188569, DOI 10.1016/0022-247X(92)90034-B
N.S. Koshliakov, Application of the theory of sum-formulae to the investigation of a class of one-valued analytical functions in the theory of numbers, Messenger Math. 58(1928/1929), 1–23.
D. Leshchiner, Some new identities for $\zeta (k)$, J. Number Theory 13 (1981), no. 3, 355–362. MR 634205, DOI 10.1016/0022-314X(81)90020-2
Wilhelm Magnus, Fritz Oberhettinger, and Raj Pal 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. MR 0232968
Srinivasa Ramanujan, Notebooks. Vols. 1, 2, Tata Institute of Fundamental Research, Bombay, 1957. MR 0099904
V. Ramaswami, Notes on Riemann’s $\zeta$-function, J. London Math. Soc. 9(1934), 165–169.
John Riordan, Combinatorial identities, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 0231725
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 (1989). MR 1007062
A. Terras, Some formulas for the Riemann zeta function at odd integer argument resulting from Fourier expansions of the Epstein zeta function, Acta Arith. 29 (1976), no. 2, 181–189. MR 396434, DOI 10.4064/aa-29-2-181-189
E. C. Titchmarsh, The theory of the Riemann zeta-function, 2nd ed., The Clarendon Press, Oxford University Press, New York, 1986. Edited and with a preface by D. R. Heath-Brown. MR 882550
Hirofumi Tsumura, On evaluation of the Dirichlet series at positive integers by $q$-calculation, J. Number Theory 48 (1994), no. 3, 383–391. MR 1293869, DOI 10.1006/jnth.1994.1074
J.R. Wilton, A proof of Burnside’s formula for $\log \Gamma ( x+1)$ and certain allied properties of Riemann’s $\zeta$-function, Messenger Math. 52(1922/1923), 90–93.
Zhang Nan Yue and Kenneth S. Williams, Some series representations of $\zeta (2n+1)$, Rocky Mountain J. Math. 23 (1993), no. 4, 1581–1592. MR 1256463, DOI 10.1216/rmjm/1181072507
Nan Yue Zhang and Kenneth S. Williams, Some infinite series involving the Riemann zeta function, Analysis, geometry and groups: a Riemann legacy volume, Hadronic Press Collect. Orig. Artic., Hadronic Press, Palm Harbor, FL, 1993, pp. 691–712. MR 1299360