On algebraic differential equations concerning the Riemann-zeta function and the Euler-gamma function

Abstract

In this paper, we prove that $\zeta$ is not a solution of any non-trivial algebraic differential equation whose coefficients are polynomials in $\Gamma ,{\Gamma }^{(n)},{\Gamma }^{(l)}$ over the ring of polynomials in $\mathbb{C}$, $l>n\ge 1$ are positive integers. We extended the result that $\zeta$ does not satisfy any non-trivial algebraic differential equation whose coefficients are polynomials in $\Gamma ,{\Gamma }^{\prime },{\Gamma }^{\prime \prime }$ over the field of complex numbers, which is proved by Li and Ye[7].

12 pages