Abstract
In this article, we study the local behaviour of the multiple zeta functions at integer points and write down a Laurent type expansion of the multiple zeta functions around these points. Such an expansion involves a convergent power series whose coefficients are obtained by a regularisation process, similar to the one used in defining the classical Stieltjes constants for the Riemann zeta function. We therefore call these coefficients multiple Stieltjes constants. The remaining part of the above mentioned Laurent type expansion is then expressed in terms of the multiple Stieltjes constants arising in smaller depths.
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Notes
Some authors, as in [3] for example, refer to the numbers \(B_k^\star \) as the Bernoulli numbers and denote them by \(B_k\).
References
Akiyama, S., Egami, S., Tanigawa, Y.: Analytic continuation of multiple zeta-functions and their values at non-positive integers. Acta Arith. 98(2), 107–116 (2001)
Akiyama, S., Tanigawa, Y.: Multiple zeta values at non-positive integers. Ramanujan J. 5, 327–351 (2001)
Arakawa, T., Ibukiyama, T., Kaneko, M.: Bernoulli Numbers and Zeta Functions (With an Appendix by Don Zagier). Springer Monographs in Mathematics, Springer, Tokyo (2014)
Baillaud, B., Bourget, H. (eds.): Correspondance d’Hermite et de Stieltjes. Tomes I, II. Gauthier-Villars, Paris (1905)
Bourbaki, N.: Commutative Algebra, Chapters 1–7. Elements of Mathematics, Springer, Berlin (1998)
Bourbaki, N.: Functions of a Real Variable. Elements of Mathematics, Springer, Berlin (2004)
Grothendieck, A.: Éléments de géométrie algébrique. I. Le langage des schémas, Inst. Hautes Études Sci. Publ. Math. 4 (1960)
Kaneko, M., Yamamoto, S.: A new integral-series identity of multiple zeta values and regularizations. Sel. Math. (N.S.) 24(3), 2499–2521 (2018)
Manchon, D., Paycha, S.: Nested sums of symbols and renormalized multiple zeta values. Int. Math. Res. Not. IMRN 2010(24), 4628–4697 (2010)
Matsumoto, K., Onozuka, T., Wakabayashi, I.: Laurent series expansions of multiple zeta-functions of Euler–Zagier type at integer points. Math. Z. 295(1–2), 623–642 (2020)
Mehta, J., Saha, B., Viswanadham, G.K.: Analytic properties of multiple zeta functions and certain weighted variants, an elementary approach. J. Number Theory 168, 487–508 (2016)
Onozuka, T.: Analytic continuation of multiple zeta-functions and the asymptotic behavior at non-positive integers. Funct. Approx. Comment. Math. 49, 331–348 (2013)
Ree, R.: Lie elements and an algebra associated with shuffles. Ann. Math. 2(68), 210–220 (1958)
Zhao, J.: Analytic continuation of multiple zeta functions. Proc. Am. Math. Soc. 128(5), 1275–1283 (2000)
Acknowledgements
I would like to record my sincere thanks to Prof. Joseph Oestelé for numerous indispensable inputs during this work and formation of this article. I am also thankful to him for introducing me to the language of asymptotic expansion of sequences of complex numbers and of germs of holomorphic and meromorphic functions, relative to a comparison scale. I am also deeply thankful to the referee for pointing out an inaccuracy in an earlier version of this article and for other insightful comments which enhanced the presentation of the article. This work was carried out in Institut de Mathématiques de Jussieu, with support from IRSES Moduli and LIA.
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Saha, B. Multiple Stieltjes constants and Laurent type expansion of the multiple zeta functions at integer points. Sel. Math. New Ser. 28, 6 (2022). https://doi.org/10.1007/s00029-021-00719-1
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DOI: https://doi.org/10.1007/s00029-021-00719-1