Multiple zeta functions and polylogarithms over global function fields

Basak, Debmalya; Degré-Pelletier, Nicolas; Lalín, Matilde N.

doi:10.5802/jtnb.1128

Basak, Debmalya ¹ ; Degré-Pelletier, Nicolas ² ; Lalín, Matilde N.²

¹ Indian Institute of Science Education and Research (IISER), Kolkata, Mohanpur, West Bengal 741246, India
² Université de Montréal, Pavillon André-Aisenstadt, Dépt. de mathématiques et de statistique, CP 6128, succ. Centre-ville Montréal, Québec, H3C 3J7, Canada

Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 403-438.

Résumé
Abstract

Dans [36], Thakur définit des analogues de la fonction zêta multiple sur les corps de fonctions, $ζ_{d} (𝔽_{q} [T]; s_{1}, \dots, s_{d})$ et $ζ_{d} (K; s_{1}, \dots, s_{d})$ , où $K$ est un corps de fonctions global. Les versions étoilées de ces fonctions ont été étudiées par Masri [28]. Nous prouvons des formules de réduction pour ces fonctions étoilées, nous définissons des analogues des polylogarithmes multiples et nous présentons quelques formules pour des valeurs zêta multiples.

In [36], Thakur defines function field analogs of the classical multiple zeta function, namely, $ζ_{d} (𝔽_{q} [T]; s_{1}, \dots, s_{d})$ and $ζ_{d} (K; s_{1}, \dots, s_{d})$ , where $K$ is a global function field. Star versions of these functions were further studied by Masri [28]. We prove reduction formulas for these star functions, extend the construction to function field analogs of multiple polylogarithms, and exhibit some formulas for multiple zeta values.

Reçu le : 2019-09-11
Accepté le : 2020-05-06
Publié le : 2020-10-29

DOI : 10.5802/jtnb.1128

Classification : 11M41, 11R58, 11T55, 14H05
Mots clés : Function field, multiple zeta function, multiple polylogarithms

Affiliations des auteurs :

Basak, Debmalya ¹ ; Degré-Pelletier, Nicolas ² ; Lalín, Matilde N. ²

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     title = {Multiple zeta functions and polylogarithms over global function fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {403--438},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
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Basak, Debmalya; Degré-Pelletier, Nicolas; Lalín, Matilde N. Multiple zeta functions and polylogarithms over global function fields. Journal de théorie des nombres de Bordeaux, Tome 32 (2020) no. 2, pp. 403-438. doi : 10.5802/jtnb.1128. http://www.numdam.org/articles/10.5802/jtnb.1128/

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