On algebraic differential equations of gamma function and Riemann zeta function

Abstract

Abstract. Due to Voronin’s universality theorem and Riemann–von Mangoldt formula, this paper concerns the problem of algebraic differential independence between the gamma function Γ and the function f(ζ), where ζ is the Riemann zeta function and f is a function with at least one zero-point. It is showed that Γ and f(ζ) cannot satisfy any nontrivial distinguished differential equation with meromorphic coefficients φ having Nevanlinna characteristic satisfying T (r, φ) = o(r) as r → ∞.