On the Difference Independence of the Euler Gamma Function and the Riemann Zeta Function

We prove that the Riemann zeta function ζ and the Euler gamma function Γ cannot satisfy a class of non-trivial algebraic difference equations with functional coefficients that are connected to the zeros of the Riemann zeta function ζ on the critical line L = { z ∈ C : Re ( z ) = 1 / 2 } . The main result of this paper is the difference analogue of the corresponding result from Li–Ye (J. Differential Equations 260(2):1456–1464, 2016), and improves the corresponding result from Chiang–Feng (Acta Arith. 125(4):317–329, 2006). Examples are provided to show that the main results in this paper, in a sense, are best possible.