Treffer: The multiplicative and functional independence of Dedekind zeta functions of abelian fields

Complex functions \(f_1,\dots,f_m\) are said to be functionally independent if for any continuous functions \(F_l:\,\,{\mathbb{C}}^m \to{\mathbb C}\,\,(l=0,\dots,q)\), not all identically zero, the function \(\sum_{l=0}^q s^lF_l(f_1(s),\dots,f_m(s))\) is not identically zero. The author proves that for Dedekind zeta-functions \(\zeta_{L_j}\) of any set of abelian fields \(L_j\) (\(j=1,\dots,m\)) this is equivalent to the multiplicative independence of these functions. The proof is based on a result by \textit{S. M. Voronin} [Acta Arith. 27, 492--503 (1975; Zbl 0308.10025)] about the functional independence of Dirichlet's L-functions. The author also shows how all multiplicative dependence relations for the functions \(\zeta_{L_j}\) can be characterized by norm relations.