In this paper, we prove that ζ is not a solution of any non-trivial algebraic differential equation whose coefficients are polynomials in ${\mathrm{\Gamma }}^{(\alpha )},{\mathrm{\Gamma }}^{(n)},{\mathrm{\Gamma }}^{(l)}$ over the ring of polynomials in $\mathbb{C}$, $l>n>\alpha \ge 0$ are nonnegative integers. We extend the result that ζ does not satisfy any non-trivial algebraic differential equation whose coefficients are polynomials in $\mathrm{\Gamma },{\mathrm{\Gamma }}^{\prime },{\mathrm{\Gamma }}^{″}$ 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.].

中文翻译：

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关于 Riemann-zeta 函数和 Euler-gamma 函数的代数微分方程

在本文中，我们证明ζ不是任何系数为多项式的非平凡代数微分方程的解${\mathrm{\Gamma }}^{(\alpha )},{\mathrm{\Gamma }}^{(n)},{\mathrm{\Gamma }}^{(l)}$在多项式环上$\mathbb{C}$,$l>n>\alpha \ge 0$是非负整数。我们扩展了ζ不满足任何系数为多项式的非平凡代数微分方程的结果$\mathrm{\Gamma },{\mathrm{\Gamma }}^{\text{'}},{\mathrm{\Gamma }}^{”}$在复数领域，这是由李和叶证明的[Li BQ，Ye Z.关于黎曼zeta函数和欧拉伽马函数的代数微分方程。印第安纳大学数学 J. 2010;59:1405–1416.]。