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Cyclotomic analogues of finite multiple zeta values

Part of: Combinatorics Algebraic number theory: global fields Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  06 November 2018

Henrik Bachmann ,
Yoshihiro Takeyama  and
Koji Tasaka
Henrik Bachmann
Affiliation:
Graduate School of Mathematics, Nagoya University, Nagoya, Aichi 464-8602, Japan email henrik.bachmann@math.nagoya-u.ac.jp
Yoshihiro Takeyama
Affiliation:
Department of Mathematics, Faculty of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan email takeyama@math.tsukuba.ac.jp
Koji Tasaka
Affiliation:
Department of Information Science and Technology, Aichi Prefectural University, Nagakute-city, Aichi 480-1198, Japan email tasaka@ist.aichi-pu.ac.jp
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Abstract

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.

Keywords

multiple zeta (star) valuesfinite multiple zeta (star) valuessymmetric multiple zeta (star) valuesKaneko–Zagier conjecturefinite multiple harmonic q-series

MSC classification

Primary: 11M32: Multiple Dirichlet series and zeta functions and multizeta values 11R18: Cyclotomic extensions 05A30: $q$-calculus and related topics
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 12 , December 2018 , pp. 2701 - 2721
Copyright
© The Authors 2018 

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