journal article
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On the Difference Independence of the Euler Gamma Function and the Riemann Zeta Function
Bibi, Amina; Li, Xiao-Min; Yi, Hong-Xun
2023 Computational Methods and Function Theory
doi: 10.1007/s40315-023-00483-7
We prove that the Riemann zeta function ζ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\zeta $$\end{document} and the Euler gamma function Γ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\Gamma $$\end{document} cannot satisfy a class of non-trivial algebraic difference equations with functional coefficients that are connected to the zeros of the Riemann zeta function ζ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\zeta $$\end{document} on the critical line L={z∈C:Re(z)=1/2}\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$L=\{z \in \mathbb {C}: {\text {Re}}(z) = 1/2\}$$\end{document}. The main result of this paper is the difference analogue of the corresponding result from Li–Ye (J. Differential Equations 260(2):1456–1464, 2016), and improves the corresponding result from Chiang–Feng (Acta Arith. 125(4):317–329, 2006). Examples are provided to show that the main results in this paper, in a sense, are best possible.