A continuous version of multiple zeta functions and multiple zeta values

Jiangtao Li

doi:10.5802/crmath.440

Number theory

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.

Abstract

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.

Received: 2022-06-27
Revised: 2022-10-25
Accepted: 2022-10-25
Published online: 2023-03-02

DOI: 10.5802/crmath.440

Classification: 11M32

Author's affiliations:

Jiangtao Li ¹

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

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

@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},
}

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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/

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