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An introduction to classical and finite multiple zeta values
Masanobu Kaneko1
1 Faculty of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2019), pp. 103-129.

We review some basic properties of multiple zeta values, in particular the theory of regularization and its connection to an identity between certain integral and series discovered in collaboration with S. Yamamoto. We also introduce the two “finite” versions of multiple zeta values, and a conjectural connection between them, which were discovered jointly with D. Zagier.

Nous décrivons certaines propriétés basiques des valeurs de fonctions zétas multiples. Nous explicitons en particulier la théorie des régularisations et son lien avec une identité, obtenue en collaboration avec S. Yamamoto, entre certaines intégrales et séries. Nous présentons également les deux versions « finies » des valeurs zétas multiples et un lien conjectural entre elles découvert conjointement avec D. Zagier.

Received:
Published online:
DOI: 10.5802/pmb.31
Classification: 11M32, 11B68
Keywords: multiple zeta values, regularization, finite multiple zeta values

Masanobu Kaneko 1

1 Faculty of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Masanobu Kaneko. An introduction to classical and finite multiple zeta values. Publications mathématiques de Besançon. Algèbre et théorie des nombres, no. 1 (2019), pp. 103-129. doi : 10.5802/pmb.31. https://pmb.centre-mersenne.org/articles/10.5802/pmb.31/

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

[2] Tsuneo Arakawa; Tomoyoshi Ibukiyama; Masanobu Kaneko Bernoulli Numbers and Zeta Functions, Springer Monographs in Mathematics, Springer, 2014 | Zbl

[3] Henrik Bachmann; Yoshihiro Takeyama; Koji Tasaka Cyclotomic analogues of finite multiple zeta values, Compos. Math., Volume 154 (2018) no. 2, pp. 2701-2721 | Zbl

[4] Pierre Deligne; Alexander B. Goncharov Groupes fondamentaux motiviques de Tate mixte, Ann. Sci. Éc. Norm. Supér., Volume 38 (2005) no. 1, pp. 1-56 | Zbl

[5] A. B. Goncharov Periods and mixed motives (2002) (preprint)

[6] Khodabakhsh Hessami Pilehrood; Tatiana Hessami Pilehrood; Roberto Tauraso New properties of multiple harmonic sums modulo p and p-analogues of Leshchiner’s series, Trans. Am. Math. Soc., Volume 366 (2014) no. 6, pp. 3131-3159 | Zbl

[7] Minoru Hirose Double shuffle relations for refined symmetric zeta values (2018) (https://arxiv.org/abs/1807.04747)

[8] Michael E. Hoffman References on multiple zeta values and Euler sums (https://www.usna.edu/Users/math/meh/biblio.html)

[9] Michael E. Hoffman The algebra of multiple harmonic series, J. Algebra, Volume 194 (1997) no. 2, pp. 477-495 | Zbl

[10] Michael E. Hoffman Quasi-symmetric functions and mod p multiple harmonic sums, Kyushu J. Math., Volume 69 (2015) no. 2, pp. 345-366 | Zbl

[11] Kentaro Ihara; Masanobu Kaneko; Don Zagier Derivation and double shuffle relations for multiple zeta values, Compos. Math., Volume 142 (2006) no. 2, pp. 307-338 | Zbl

[12] Kohtaro Imatomi; Masanobu Kaneko; Erika Takeda Multi-poly-Bernoulli numbers and finite multiple zeta values, J. Integer Seq., Volume 17 (2014) no. 4, 14.4.5, 12 pages | Zbl

[13] David Jarossay Double mélange des multizêtas finis et multizêtas symétrisés, C. R. Math. Acad. Sci. Paris, Volume 352 (2014) no. 10, pp. 767-771 | Zbl

[14] Ken Kamano Finite Mordell-Tornheim multiple zeta values, Funct. Approximatio, Comment. Math., Volume 54 (2015) no. 1, pp. 65-72 | Zbl

[15] Ken Kamano Weighted sum formulas for finite multiple zeta values, J. Number Theory, Volume 192 (2018), pp. 168-180 | Zbl

[16] Masanobu Kaneko; Kojiro Oyama; Shingo Saito Analogues of Aoki–Ohno and Le–Murakami relations in finite multiple zeta values (2018) (https://arxiv.org/abs/1810.04813)

[17] Masanobu Kaneko; Shuji Yamamoto A new integral-series identity of multiple zeta values and regularizations, Sel. Math., New Ser., Volume 24 (2018) no. 3, pp. 2499-2521 | Zbl

[18] Masanobu Kaneko; Don Zagier Finite multiple zeta values (in preparation)

[19] Kohji Matsumoto On the analytic continuation of various multiple zeta-functions, Number theory for the millennium II, A K Peters, 2002, pp. 417-440 | Zbl

[20] Hideki Murahara Derivation relations for finite multiple zeta values, Int. J. Number Theory, Volume 13 (2017) no. 2, pp. 419-427 | Zbl

[21] Hideki Murahara; Mika Sakata On multiple zeta values and finite multiple zeta values of maximal height, Int. J. Number Theory, Volume 14 (2018) no. 4, pp. 975-987 | Zbl

[22] Masataka Ono Finite multiple zeta values associated with 2-colored rooted trees, J. Number Theory, Volume 181 (2017), pp. 99-116 | Zbl

[23] Kojiro Oyama Ohno-type relation for finite multiple zeta values, Kyushu J. Math., Volume 72 (2018) no. 2, pp. 277-285 | Zbl

[24] Christophe Reutenauer Free Lie Algebras, London Mathematical Society Monographs, 7, Clarendon Press, 1993 | Zbl

[25] Julian Rosen Multiple harmonic sums and Wolstenholme’s theorem, Int. J. Number Theory, Volume 9 (2013) no. 8, pp. 2033-2052 | Zbl

[26] Shingo Saito; Noriko Wakabayashi Sum formula for finite multiple zeta values, J. Math. Soc. Japan, Volume 67 (2015) no. 3, pp. 1069-1076 | Zbl

[27] Shingo Saito; Noriko Wakabayashi Bowman-Bradley type theorem for finite multiple zeta values, Tôhoku Math. J., Volume 68 (2016) no. 2, pp. 241-251 | Zbl

[28] Kenji Sakugawa; Shin-ichiro Seki Finite and etale polylogarithms, J. Number Theory, Volume 176 (2017), pp. 279-301 | Zbl

[29] Kenji Sakugawa; Shin-ichiro Seki On functional equations of finite multiple polylogarithms, J. Algebra, Volume 469 (2017), pp. 323-357 | Zbl

[30] W. Specht Die linearen Beziehungen zwischen höheren Kommutatoren, Math. Z., Volume 51 (1948), pp. 367-376 | Zbl

[31] Roberto Tauraso Congruences involving alternating multiple harmonic sums, Electron. J. Comb., Volume 17 (2010) no. 1, R16, 11 pages | Zbl

[32] Roberto Tauraso; Jianqiang Zhao Congruences of alternating multiple harmonic sums, J. Comb. Number Theory, Volume 2 (2010) no. 2, pp. 129-159 | Zbl

[33] Tomohide Terasoma Mixed Tate motives and multiple zeta values, Invent. Math., Volume 149 (2002) no. 2, pp. 339-369 | Zbl

[34] Harry S. Vandiver On developments in an arithmetic theory of the Bernoulli and allied numbers, Scripta Math., Volume 25 (1961), pp. 273-303 | Zbl

[35] Michel Waldschmidt Valeurs zêta multiples. Une introduction, J. Théor. Nombres Bordx, Volume 12 (2000) no. 2, pp. 581-595 | Zbl

[36] Shuji Yamamoto Multiple zeta-star values and multiple integrals, RIMS Kôkyûroku Bessatsu, Volume B68 (2017), pp. 3-14 | Zbl

[37] Seidai Yasuda Finite real multiple zeta values generate the whole space Z, Int. J. Number Theory, Volume 12 (2016) no. 3, pp. 787-812 | Zbl

[38] Don Zagier Values of zeta functions and their applications, First European congress of mathematics (ECM) (Progress in Mathematics), Volume 120, Birkhäuser, 1994, pp. 497-512 | Zbl

[39] Jianqiang Zhao Wolstenholme type theorem for multiple harmonic sums, Int. J. Number Theory, Volume 4 (2008) no. 1, pp. 73-106 | Zbl

[40] Jianqiang Zhao Mod p structure of alternating and non-alternating multiple harmonic sums, J. Théor. Nombres Bordx, Volume 23 (2011) no. 1, pp. 299-308 | Zbl

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