PreprintCC BY 4.0arXiv
Formal finite multiple zeta values
Abstract
We study the algebra of formal finite multiple zeta values by imposing the stuffle and linear shuffle relations of Kaneko and Zagier. Our first results are a surjective homomorphism to the algebra of formal symmetric multiple zeta values and a parity reduction in depth at most four. In even weight we construct a quotient of the formal double zeta space which is isomorphic to the space of formal finite multiple zeta values of depth at most four. The proof combines an explicit depth-three reduction with a motivic lower bound. We also attach to every even period polynomial an explicit relation among the values ζA(2a,1,2b,1), and these give all relations among the corresponding formal values.
Related
- A generating function identity for finite and symmetric multiple zeta values via Dougall's bilateral 2H2 summation
- Refined sum formulas for finite multiple zeta values of level two
- A Unified Approach to Formal Arakawa-Kaneko Type Zeta Values via Iterated Integrals
- Multiple Zeta Values
- New Hoffman-type generators for alternating multiple zeta values
BibTeX
@misc{bachmann2026formal,
title = {{Formal finite multiple zeta values}},
author = {Henrik Bachmann and Risan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2608.15480},
eprint = {2608.15480},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.15480},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Formal finite multiple zeta values
AU - Henrik Bachmann
AU - Risan
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2608.15480
UR - https://arxiv.org/abs/2608.15480
AB - We study the algebra of formal finite multiple zeta values by imposing the stuffle and linear shuffle relations of Kaneko and Zagier. Our first results are a surjective homomorphism to the algebra of formal symmetric multiple zeta values and a parity reduction in depth at most four. In even weight we construct a quotient of the formal double zeta space which is isomorphic to the space of formal finite multiple zeta values of depth at most four. The proof combines an explicit depth-three reduction with a motivic lower bound. We also attach to every even period polynomial an explicit relation among the values ζ_A(2a,1,2b,1), and these give all relations among the corresponding formal values.
ER -