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

Qi Han; Jingbo Liu

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arXiv · 1811.04188

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

Qi Han·Jingbo Liu

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Abstract

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

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Qi Han, Jingbo Liu. 2018-11-10. Algebraic differential independence regarding the Riemann $\boldsymbol{\zeta}$-function and the Euler $\boldsymbol{\Gamma}$-function. https://arxiv.org/abs/1811.04188

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