Non-vanishing of finite multiple zeta values in A

Determine whether there exists a tuple of positive integers (k_1, ..., k_r) such that the finite multiple zeta value ζ_A(k_1, ..., k_r) ∈ A = (∏_{p prime} Z/pZ)/(⊕_{p prime} Z/pZ), defined by ζ_A(k_1, ..., k_r) = (∑_{0 < m_1 < ⋯ < m_r < p} 1/(m_1^{k_1} ⋯ m_r^{k_r}) mod p)_p, is nonzero.

Background

The paper reviews the construction of the ring A and the finite multiple zeta values (FMZVs) introduced by Kaneko and Zagier, defined as reductions modulo primes p assembled into an element of A. Despite structural parallels with classical multiple zeta values, basic arithmetic properties of FMZVs in A remain elusive.

The authors highlight that even establishing the existence of any nonzero FMZV is currently beyond reach, underscoring a fundamental gap in the emerging arithmetic of A.

References

In fact, at present, it is not known whether there exists a tuple $(k_1, \dots, k_r)$ such that $\zeta_A(k_1, \dots, k_r) \neq 0$.

— On finite analogues of Dobiński's formula and of Euler's constant via Gregory polynomials  (2604.01578 - Matsusaka et al., 2 Apr 2026) in Section 1 (Introduction)

However, the following seemingly naive-looking, yet fundamental, and in fact very difficult question does not have a definite answer yet: Is there a nonzero element in $FMZ$?

— Effective Euler Sums  (2609.09260 - Zhao, 8 Sep 2026) in Section 1.3, immediately before the subsection “Finite Euler sum and the key conjecture”