Differential independence of Γ and ζ

In a celebrated theorem Holder proved that the Euder Gamma-function is differential transcendental, i.e. Gamma(z) is not a solution of any (non-trivial) algebraic ordinary differential equation with coefficients that are complex numbers; and we extend his methods to the Riemann zeta-function. Moreover, we conjecture that Gamma and zeta are differential independent, i.e. Gamma(z) is not a solution of any such algebraic differential equation-even allowing coefficients that are differential polynomials in zeta(z). However, we are able to demonstrate only the partial result that Gamma(z) and zeta(sin 2 pi z) are differential independent.