# Rapidly convergent series representations for ζ(2n+1) and their χ-analogue

Acta Arithmetica (1999)

- Volume: 90, Issue: 1, page 79-89
- ISSN: 0065-1036

## Access Full Article

top## How to cite

topMasanori Katsurada. "Rapidly convergent series representations for ζ(2n+1) and their χ-analogue." Acta Arithmetica 90.1 (1999): 79-89. <http://eudml.org/doc/207316>.

@article{MasanoriKatsurada1999,

author = {Masanori Katsurada},

journal = {Acta Arithmetica},

keywords = {Riemann zeta-function; Dirichlet L-function; Mellin-Barnes integral; series representation; Dirichlet -functions; asymptotic expansion; Bernoulli numbers; series representations},

language = {eng},

number = {1},

pages = {79-89},

title = {Rapidly convergent series representations for ζ(2n+1) and their χ-analogue},

url = {http://eudml.org/doc/207316},

volume = {90},

year = {1999},

}

TY - JOUR

AU - Masanori Katsurada

TI - Rapidly convergent series representations for ζ(2n+1) and their χ-analogue

JO - Acta Arithmetica

PY - 1999

VL - 90

IS - 1

SP - 79

EP - 89

LA - eng

KW - Riemann zeta-function; Dirichlet L-function; Mellin-Barnes integral; series representation; Dirichlet -functions; asymptotic expansion; Bernoulli numbers; series representations

UR - http://eudml.org/doc/207316

ER -

## References

top- [Ay] R. Ayoub, Euler and the zeta function, Amer. Math. Monthly 81 (1974), 1067-1086. Zbl0293.10001
- [Ch] B. R. Choe, An elementary proof of ${\sum}_{n=1}^{\infty}1/n\xb2={\pi}^{2}/6$, Amer. Math. Monthly 94 (1987), 662-663. Zbl0624.40001
- [CK] D. Cvijović and J. Klinowski, New rapidly convergent series representations for ζ(2n+1), Proc. Amer. Math. Soc. 125 (1997), 1263-1271. Zbl0863.11055
- [Ew1] J. A. Ewell, A new series representation for ζ(3), Amer. Math. Monthly 97 (1990), 219-220.
- [Ew2] J. A. Ewell, On values of the Riemann zeta function at integral arguments, Canad. Math. Bull. 34 (1991), 60-66.
- [Ew3] J. A. Ewell, On the zeta function values ζ(2k+1), k=1,2,..., Rocky Mountain J. Math. 23 (1995), 1003-1012.
- [Iv] A. Ivić, The Riemann Zeta-Function, Wiley, New York, 1985.
- [Ka1] M. Katsurada, Power series with the Riemann zeta-function in the coefficients, Proc. Japan Acad. Ser. A 72 (1996), 61-63. Zbl0860.11050
- [Ka2] M. Katsurada, On Mellin-Barnes type of integrals and sums associated with the Riemann zeta-function, Publ. Inst. Math. (Beograd) (N.S.) 62 (76) (1997), 13-25.
- [Ka3] M. Katsurada, Power series and asymptotic series associated with the Lerch zeta-function, Proc. Japan Acad. Ser. A 74 (1998), 167-170. Zbl0937.11035
- [Ra] V. Ramaswami, Notes on Riemann's ζ-function, J. London Math. Soc. 9 (1934), 165-169. Zbl0009.34801
- [Sr1] 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. Zbl0659.10047
- [Sr2] H. M. Srivastava, Certain families of rapidly convergent series representations for ζ(2n+1), Math. Sci. Research Hot-Line 1 (6) (1997), 1-6.
- [Sr3] H. M. Srivastava, Some rapidly converging series for ζ(2n+1), Proc. Amer. Math. Soc. 127 (1999), 385-396. Zbl0903.11020
- [Ti] E. C. Titchmarsh, The Theory of the Riemann Zeta-function, 2nd ed., revised by D. R. Heath-Brown, Oxford Univ. Press, 1986.
- [Wa] L. C. Washington, Introduction to Cyclotomic Fields, Springer, New York, 1982. Zbl0484.12001
- [Wi] J. R. Wilton, A proof of Burnside's formula for Γ(x+1) and certain allied properties of Riemann's ζ-function, Messenger Math. 52 (1922/1923), 90-93.
- [YW] Z.-N. Yue and K. S. Williams, Some series representations of ζ(2n+1), Rocky Mountain J. Math. 23 (1993), 1581-1591.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.