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

Abstract

In this paper, we prove that ζ is not a solution of any non-trivial algebraic differential equation whose coefficients are polynomials in over the ring of polynomials in , are nonnegative integers. We extend the result that ζ does not satisfy any non-trivial algebraic differential equation whose coefficients are polynomials in over the field of complex numbers, which is proved by Li and Ye [Li BQ, Ye Z. Algebraic differential equations concerning the Riemann zeta function and the Euler gamma function. Indiana Univ Math J. 2010;59:1405–1416.].