Zagier’s conjecture on expressing Dedekind zeta values via polylogarithms

Determine whether, for any number field F and integer n ≥ 2, the special value ζ_F(n) can be expressed in terms of special values of the classical polylogarithm Li_n evaluated at elements of F; that is, express ζ_F(n) through Q-linear combinations of Li_n(x) with x ∈ F.

Background

Borel’s theorem relates ζF(n) to higher regulators on K{2n−1}(F), but explicit computations are difficult. The conjecture proposes a concrete description of ζ_F(n) via polylogarithmic special values, linking L-values to explicit periods.

Such expressions would connect the analytic behavior of Dedekind zeta functions to the geometry of mixed Tate motives and the arithmetic of number fields.