Differential and difference independence of \(\zeta\) and \(\Gamma\) (Q6644985)

Let \(\Gamma\) be the Euler gamma function and \(\zeta\) denotes the Riemann zeta function.\N\NIn this paper under review, the authors prove mainly (see Theorem 1) that if \(P\) be a polynomial of \(m + n + 1\) variables in \(\mathbb{C}[X,Y_{0},\dots,Y_{m+n-1}]\) and \N\[\NP(s,\zeta(s),\dots,\zeta^{(m-1)}(s), \Gamma(s),\dots,\Gamma^{(n-1)}(s)) \equiv 0,\N\] \Nthen \(P\equiv 0\). In other words, \(\zeta\) cannot satisfy any non-trivial algebraic differential equation whose coefficients are differential polynomials of \(\Gamma\) (known as Markus's Conjecture [\textit{L. Markus}, J. Dyn. Differ. Equations 19, No. 1, 133--154 (2007; Zbl 1119.33004)]). Moreover, in Theorem 2 they state the difference analogue of Markus's Conjecture.