Comptes Rendus
Number theory
Multiple zeta values at the non-positive integers
Comptes Rendus. Mathématique, Volume 352 (2014) no. 12, pp. 977-984.

In this paper, we provide an alternative method to calculate the multiple zeta values at non-positive integers by means of Raabe's formula and the Bernoulli numbers.

Dans cet article, nous proposons une autre méthode pour calculer les valeurs de la fonction zêta multiple aux entiers négatifs à l'aide de la formule de Raabe et des nombres de Bernoulli.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2014.10.001

Boualem Sadaoui 1

1 Université de Khemis Miliana, Laboratoire LESI, 44225, Khemis Miliana, Algeria
@article{CRMATH_2014__352_12_977_0,
     author = {Boualem Sadaoui},
     title = {Multiple zeta values at the non-positive integers},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {977--984},
     publisher = {Elsevier},
     volume = {352},
     number = {12},
     year = {2014},
     doi = {10.1016/j.crma.2014.10.001},
     language = {en},
}
TY  - JOUR
AU  - Boualem Sadaoui
TI  - Multiple zeta values at the non-positive integers
JO  - Comptes Rendus. Mathématique
PY  - 2014
SP  - 977
EP  - 984
VL  - 352
IS  - 12
PB  - Elsevier
DO  - 10.1016/j.crma.2014.10.001
LA  - en
ID  - CRMATH_2014__352_12_977_0
ER  - 
%0 Journal Article
%A Boualem Sadaoui
%T Multiple zeta values at the non-positive integers
%J Comptes Rendus. Mathématique
%D 2014
%P 977-984
%V 352
%N 12
%I Elsevier
%R 10.1016/j.crma.2014.10.001
%G en
%F CRMATH_2014__352_12_977_0
Boualem Sadaoui. Multiple zeta values at the non-positive integers. Comptes Rendus. Mathématique, Volume 352 (2014) no. 12, pp. 977-984. doi : 10.1016/j.crma.2014.10.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.10.001/

[1] S. Akiyama; S. Egami; Y. Tanigawa Analytic continuation of multiple zeta-functions and their values at non-positive integers, Acta Arith., Volume 98 (2001) no. 2, pp. 107-116

[2] T.M. Apostol Introduction to Analytic Number Theory, Springer, 1976

[3] F.V. Atkinson The mean-value of the Riemann zeta function, Acta Math., Volume 81 (1949), pp. 353-376

[4] J.M. Borwein; D.M. Bradley; D.J. Broadhurst; P. Lisoněk Combinatorial aspects of multiple zeta values, Electron. J. Comb., Volume 5 (1998), pp. 1077-8926

[5] L. Euler Meditationes circa singulare serierum genus, Novi Commun. Acad. Sci. Petropol., Volume 20 (1775), pp. 140-186 (reprinted in Opera Omnia, Ser. I, vol. 15, B.G. Teubner, Berlin, 1927, pp. 217–267)

[6] E. Friedman; A. Pereira Special values of Dirichlet series and zeta integrals, Int. J. Number Theory, Volume 8 (2012) no. 3, pp. 697-714

[7] E. Friedman; S. Ruijsenaars Shintani–Barnes zeta and gamma functions, Adv. Math., Volume 187 (2004), pp. 362-395

[8] A. Granville A decomposition of Riemann's Zeta-function (Y. Motohashi, ed.), Analytic Number Theory, London Mathematical Society Lecture Note Series, vol. 247, Cambridge University Press, 1997, pp. 95-101

[9] L. Guo; B. Zhang Renormalization of multiple zeta values, J. Algebra, Volume 319 (2008) no. 9, pp. 3770-3809

[10] M. Hoffman Multiple harmonic series, Pac. J. Math., Volume 152 (1992), pp. 257-290

[11] D. Manchon; S. Paycha Nested sums of symbols and renormalised multiple zeta functions, Int. Math. Res. Not., Volume 24 (2010), pp. 4628-4697

[12] K. Matsumoto Analytic continuation of multiple zeta functions, Proc. Amer. Math. Soc., Volume 128 (2003), pp. 223-243

[13] Y. Ohno A generalization of the duality and sum formula on the multiple zeta values, J. Number Theory, Volume 101 (1999), pp. 39-43

[14] D. Zagier Periods of modular forms, traces of Hecke operators and multiple zeta values, Kyoto, 1992, Volume vol. 843 (1993), pp. 162-170

[15] D. Zagier Values of zeta functions and their applications (A. Joseph et al., eds.), First European Congress of Mathematics (Paris, 1992), Vol. II, Birkhäuser, Basel, 1994, pp. 497-512

[16] J.Q. Zhao Analytic continuation of multiple zeta functions, Proc. Amer. Math. Soc., Volume 128 (2000), pp. 1275-1283

Cited by Sources:

Comments - Policy