A continuous version of multiple zeta functions and multiple zeta values

Jiangtao Li

doi:10.5802/crmath.440

Théorie des nombres

A continuous version of multiple zeta functions and multiple zeta values

Jiangtao Li ¹

¹ School of Mathematics and Statistics, HNP-LAMA, Central South University, Hunan Province, China

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 697-713.

Résumé

In this paper we define a continuous version of multiple zeta functions. They can be analytically continued to meromorphic functions on $ℂ^{r}$ with only simple poles at some special hyperplanes. The evaluations of these functions at positive integers (continuous multiple zeta values) satisfy the shuffle product. We give a detailed analysis about the depth structure of continuous multiple zeta values. There are also sum formulas for continuous multiple zeta values. Lastly we calculate some special continuous multiple zeta values in terms of special values of multiple polylogarithms.

Reçu le : 2022-06-27
Révisé le : 2022-10-25
Accepté le : 2022-10-25
Publié le : 2023-03-02

DOI : 10.5802/crmath.440

Classification : 11M32

Affiliations des auteurs :

Jiangtao Li ¹

¹ School of Mathematics and Statistics, HNP-LAMA, Central South University, Hunan Province, China

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2023__361_G3_697_0,
     author = {Jiangtao Li},
     title = {A continuous version of multiple zeta functions and multiple zeta values},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {697--713},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.440},
     language = {en},
}

TY  - JOUR
AU  - Jiangtao Li
TI  - A continuous version of multiple zeta functions and multiple zeta values
JO  - Comptes Rendus. Mathématique
PY  - 2023
SP  - 697
EP  - 713
VL  - 361
PB  - Académie des sciences, Paris
DO  - 10.5802/crmath.440
LA  - en
ID  - CRMATH_2023__361_G3_697_0
ER  -

%0 Journal Article
%A Jiangtao Li
%T A continuous version of multiple zeta functions and multiple zeta values
%J Comptes Rendus. Mathématique
%D 2023
%P 697-713
%V 361
%I Académie des sciences, Paris
%R 10.5802/crmath.440
%G en
%F CRMATH_2023__361_G3_697_0

Jiangtao Li. A continuous version of multiple zeta functions and multiple zeta values. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 697-713. doi : 10.5802/crmath.440. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.440/

Bibliographie
Cité par

[1] David J. Broadhurst; Dirk Kreimer Association of multiple zeta values with positive knots via Feynman diagrams up to $9$ loops, Phys. Lett., B, Volume 393 (1997) no. 3-4, pp. 403-412 | DOI | MR | Zbl

[2] Kuo-Tsai Chen Iterated path integrals, Bull. Am. Math. Soc., Volume 83 (1977), pp. 831-879 | DOI | MR | Zbl

[3] Herbert Gangl; Masanobu Kaneko; Don Zagier Double zeta values and modular forms, Automorphic forms and zeta functions, World Scientific, 2006, pp. 71-106 | DOI | Zbl

[4] Alexander B. Goncharov Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math., Volume 114 (1995) no. 2, pp. 197-318 | DOI | MR | Zbl

[5] Alexander B. Goncharov The dihedral Lie algebras and Galois symmetries of $π_{1}^{(l)} (ℙ^{1} - {0, \infty} \cup μ_{N})$ , Duke Math. J., Volume 110 (2001) no. 3, pp. 397-487 | DOI | MR

[6] Alexander B. Goncharov; Daniil Rudenko Motivic correlators, cluster varieties, and Zagier’s conjecture on $ζ_{F} (4)$ (2018) (https://arxiv.org/abs/1803.08585)

[7] 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 | DOI | MR | Zbl

[8] Maxim Kontsevich; Don Zagier Periods, Mathematics unlimited—2001 and beyond, Springer, 2001, pp. 771-808 | DOI | Zbl

[9] Shuji Yamamoto Multiple zeta-star values and multiple integrals (2017) (https://arxiv.org/abs/1405.6499)

[10] Don Zagier Hyperbolic manifolds and special values of Dedekind zeta functions, Invent. Math., Volume 83 (1986), pp. 285-301 | DOI | MR | Zbl

[11] Don Zagier Polylogarithms, zeta-functions, and algebraic K-theory of fields, Arithmetic algebraic geometry (Progress in Mathematics), Volume 89, Birkhäuser, 1991, pp. 392-430 | MR | Zbl

[12] Jianqiang Zhao Analytic continuation of multiple zeta functions, Proc. Am. Math. Soc., Volume 128 (1999) no. 5, pp. 1275-1283 | DOI | MR | Zbl

Cité par Sources :

Commentaires - Politique