Joint Functional Independence of the Riemann Zeta-Function

Abstract

By the Ostrowski theorem, the Riemann zeta-function \(\zeta (s)\) does not satisfy any algebraic-differential equation. Voronin proved that the function \(\zeta (s)\) does not satisfy algebraic-differential equation with continuous coefficients. In the paper, a joint generalization of the Voronin theorem is given, i. e., that a collection \((\zeta (s_1), \dots , \zeta (s_r))\) does not satisfy a certain algebraic-differential equation with continuous coefficients.