Cyclotomic analogues of finite multiple zeta values

We study the values of finite multiple harmonic $q$ -series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetric multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we prove the duality formu...

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Bibliographic Details
Published inCompositio mathematica Vol. 154; no. 12; pp. 2701 - 2721
Main Authors Bachmann, Henrik, Takeyama, Yoshihiro, Tasaka, Koji
Format Journal Article
LanguageEnglish
Published London Cambridge University Press 01.12.2018
Subjects
Series (mathematics)
Unity
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Summary:We study the values of finite multiple harmonic $q$ -series at a primitive root of unity and show that these specialize to the finite multiple zeta value (FMZV) and the symmetric multiple zeta value (SMZV) through an algebraic and analytic operation, respectively. Further, we prove the duality formula for these values, as an example of linear relations, which induce those among FMZVs and SMZVs simultaneously. This gives evidence towards a conjecture of Kaneko and Zagier relating FMZVs and SMZVs. Motivated by the above results, we define cyclotomic analogues of FMZVs, which conjecturally generate a vector space of the same dimension as that spanned by the finite multiple harmonic $q$ -series at a primitive root of unity of sufficiently large degree.
ISSN:0010-437X
1570-5846
DOI:10.1112/S0010437X18007583