Establish generalized Kaneko–Zagier–type relations for p-adic finite multiple zeta values

Establish that p-adic finite multiple zeta values satisfy relations of the same form as those satisfied by t-adic symmetric multiple zeta values, as predicted by the generalized Kaneko–Zagier conjecture. Concretely, prove the p-adic finite multiple zeta value analogs of the algebraic relations known for t-adic symmetric multiple zeta values that the conjecture asserts should coincide in form.

Background

The paper proves a conjectured formula expressing certain t-adic symmetric multiple zeta values for indices ({1,3}^n) as polynomials in Riemann zeta values modulo t^2. The authors then discuss broader implications suggested by the generalized Kaneko–Zagier conjecture.

This conjecture predicts that t-adic symmetric multiple zeta values (t-adic SMZVs) and p-adic finite multiple zeta values (FMZVs) obey relations of the same form. While the paper confirms specific t-adic relations, the corresponding relations for p-adic FMZVs have not yet been established. Demonstrating these would extend the t-adic results to the p-adic setting.