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In this paper, we prove that $\boldsymbol{\zeta}$ cannot be a solution to anynontrivial algebraic differential equation whose coefficients are polynomialsin $\boldsymbol{\Gamma},\boldsymbol{\Gamma}^{(n)}$ and$\boldsymbol{\Gamma}^{(\ell n)}$ over the ring of polynomials in $\mathbf{C}$,where $\ell,n\geq1$ are positive integers.

Algebraic differential independence regarding the Riemann $\boldsymbol{\zeta}$-function and the Euler $\boldsymbol{\Gamma}$-function