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 ${\zeta }_A(2a,1,2b,1)$, and these give all relations among the corresponding formal values.

35 pages. Update in v2: Former conjecture proved, giving the formal depth-four isomorphism in every even weight