Accelerated Series for Riemann Zeta Function at Odd Integer Arguments
Affiliation(s)
VTT Technical Research Centre of Finland, Espoo, Finland. Department of Applied Physics, University of Eastern Finland, Kuopio, Finland.
VTT Technical Research Centre of Finland, Espoo, Finland. Department of Applied Physics, University of Eastern Finland, Kuopio, Finland.
ABSTRACT
Riemann zeta function is an important tool in signal analysis and number theory. Applications of the zeta function include e.g. the generation of irrational and prime numbers. In this work we present a new accelerated series for Riemann zeta function. As an application we describe the recursive algorithm for computation of the zeta function at odd integer arguments.
Riemann zeta function is an important tool in signal analysis and number theory. Applications of the zeta function include e.g. the generation of irrational and prime numbers. In this work we present a new accelerated series for Riemann zeta function. As an application we describe the recursive algorithm for computation of the zeta function at odd integer arguments.
Cite this paper
J. Olkkonen and H. Olkkonen, "Accelerated Series for Riemann Zeta Function at Odd Integer Arguments," Open Journal of Discrete Mathematics, Vol. 3 No. 1, 2013, pp. 18-20. doi: 10.4236/ojdm.2013.31004.
J. Olkkonen and H. Olkkonen, "Accelerated Series for Riemann Zeta Function at Odd Integer Arguments," Open Journal of Discrete Mathematics, Vol. 3 No. 1, 2013, pp. 18-20. doi: 10.4236/ojdm.2013.31004.
References
[1] J. M. Borwein, D. M. Bradley and R. E. Crandall, “Computational Strategies for the Riemann Zeta Function,” Journal of Computational and Applied Mathematics, Vol. 121, No. 1-2, 2000, pp. 247-296. doi:10.1016/S0377-0427(00)00336-8 [2] E. Grosswald, “Remarks Concerning the Values of the Riemann Zeta Function at Integral, Odd Arguments,” Journal of Number Theory, Vol. 4, No. 3, 1972, pp. 225-235. doi:10.1016/0022-314X(72)90049-2 [3] D. Cvijovi? and J. Klinowski, “Integral Representations of the Riemann Zeta Function for Odd-Integer Arguments,” Journal of Computational and Applied Mathematics, Vol. 142, No. 2, 2002, pp. 435-439. doi:10.1016/S0377-0427(02)00358-8 [4] T. Ito, “On an Integral Representation of Special Values of the Zeta Function at Odd Integers,” Journal of Mathematical Society of Japan, Vol. 58, No. 3, 2006, pp. 681-691. doi:10.2969/jmsj/1156342033 [5] H. Olkkonen and J. T. Olkkonen, “Fast Converging Series for Riemann Zeta Function,” Open Journal of Discrete Mathematics, Vol. 2, No. 4, 2012, pp. 131-133. doi:10.4236/ojdm.2012.24025 [6] R. Apéry, “Irrationalité de ζ(2) et ζ(3),” Astérisque, Vol. 61, 1979, pp. 11-13. [7] F. Beukers, “A Note on the Irrationality of ζ(2) and ζ(3),” Bulletin London Mathematical Society, Vol. 11, No. 3, 1979, pp. 268-272. doi:10.1112/blms/11.3.268 [8] H. Olkkonen and J. T. Olkkonen, “Log-Time Sampling of Signals: Zeta Transform,” Open Journal of Discrete Mathematics, Vol. 1, No. 2, 2011, pp. 62-65. doi:10.4236/ojdm.2011.12008
[1] J. M. Borwein, D. M. Bradley and R. E. Crandall, “Computational Strategies for the Riemann Zeta Function,” Journal of Computational and Applied Mathematics, Vol. 121, No. 1-2, 2000, pp. 247-296. doi:10.1016/S0377-0427(00)00336-8 [2] E. Grosswald, “Remarks Concerning the Values of the Riemann Zeta Function at Integral, Odd Arguments,” Journal of Number Theory, Vol. 4, No. 3, 1972, pp. 225-235. doi:10.1016/0022-314X(72)90049-2 [3] D. Cvijovi? and J. Klinowski, “Integral Representations of the Riemann Zeta Function for Odd-Integer Arguments,” Journal of Computational and Applied Mathematics, Vol. 142, No. 2, 2002, pp. 435-439. doi:10.1016/S0377-0427(02)00358-8 [4] T. Ito, “On an Integral Representation of Special Values of the Zeta Function at Odd Integers,” Journal of Mathematical Society of Japan, Vol. 58, No. 3, 2006, pp. 681-691. doi:10.2969/jmsj/1156342033 [5] H. Olkkonen and J. T. Olkkonen, “Fast Converging Series for Riemann Zeta Function,” Open Journal of Discrete Mathematics, Vol. 2, No. 4, 2012, pp. 131-133. doi:10.4236/ojdm.2012.24025 [6] R. Apéry, “Irrationalité de ζ(2) et ζ(3),” Astérisque, Vol. 61, 1979, pp. 11-13. [7] F. Beukers, “A Note on the Irrationality of ζ(2) and ζ(3),” Bulletin London Mathematical Society, Vol. 11, No. 3, 1979, pp. 268-272. doi:10.1112/blms/11.3.268 [8] H. Olkkonen and J. T. Olkkonen, “Log-Time Sampling of Signals: Zeta Transform,” Open Journal of Discrete Mathematics, Vol. 1, No. 2, 2011, pp. 62-65. doi:10.4236/ojdm.2011.12008