Establish 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 suggested by the generalized Kaneko–Zagier conjecture, thereby proving these relations for the p-adic finite setting.

Background

The paper proves a conjectured polynomial expression modulo t2 for t-adic symmetric multiple zeta values associated with indices ({1,3}n), extending prior work on such structures and showing that these values can be expressed in terms of Riemann zeta values.

Motivated by the generalized Kaneko–Zagier conjecture, the authors note that analogous relations are expected to hold for p-adic finite multiple zeta values. However, while the t-adic symmetric multiple zeta values satisfy certain relations, it remains unproven that p-adic finite multiple zeta values obey relations of the same form.

References

This idea comes from the generalized Kaneko-Zagier conjecture, which suggests that t-adic SMZVs and p-adic finite MZVs satisfy relations of the same form. Note that these relations for p-adic finite MZVs have not been proved yet.

— $t$-adic symmetric multiple zeta values for indices with alternating $1$ and $3$, starting with $1$ and ending with $3$  (2503.08380 - Fujita, 11 Mar 2025) in Section 1 (Introduction)

By using their own heuristic definition of effective double zeta values $Z_A(s,w-s)$ in Definitionf~\ref{defn:KZdefn}, Kaneko and Zagier made the following conjecture. Let $a,b\inN$ such that $w=a+b\ge 4$ is even. Set \Psi_s:=\zeta_A(s,1,w-s-1)-Z_A(s)Z_A(w-s)\quad (1\le s\le w-1). Then

— Effective Euler Sums  (2609.09260 - Zhao, 8 Sep 2026) in Conjecture 3.1, Section 5, “Explicit Effective Double Euler Sums of Even Weight”