Cyclotomic analogues of finite multiple zeta values. (6th December 2018)

• Record Type: Journal Article
• Authors: Bachmann, Henrik; Takeyama, Yoshihiro; Tasaka, Koji
• Is Part Of: Compositio mathematica. Volume 154:Number 12(2018)
• Page Start: 2701
• Page End: 2721
• Publication Date: 2018-12-06
• Subjects: 11M32, -- 11R18, -- 05A30 (primary); multiple zeta (star) values, -- finite multiple zeta (star) values, -- symmetric multiple zeta (star) values, -- Kaneko–Zagier conjecture, -- finite multiple harmonic q-series; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X18007583
• Languages: English
• ISSNs: 0010-437X

Abstract

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.