On the Amick-Fraenkel conjecture (Q2922420)

Let \(S\) be the set consisting of the number 1 and all the positive roots of the equation \(\sqrt{3}(1+ z)= \tan(z\pi/2)\). \textit{C. J. Amick} and \textit{L. E. Fraenkel} [Trans. Am. Math. Soc. 299, 273--98 (1987; Zbl 0617.76015)] conjectured that the set \(S\) is linearly independent over \(\mathbb{Q}\).NEWLINENEWLINE In the present paper, the author considers generally the equation \(\beta^z= (az+ b)/(cz+ d)\), where \(a\), \(b\), \(c\), \(d\), \(\beta\) are algebraic numbers, \(ad- bc\neq 0\), \(\beta\neq 0,1\), \(\beta^z:= e^{z\ln\beta}\) (the equation \(\sqrt{3}(1+ z)= \tan(z\pi/2)\) is its special example), and he proves that if \(z_1,\dots, z_n\in\mathbb{C}\setminus\mathbb{Q}\) are its distinct roots, and the numbers \(1\), \(z_j\), \(z_k\) are linearly independent over \(\mathbb{Q}\) for every \(j\neq k\), then Schanuel's conjecture implies the algebraic independence of \(z_1,\dots, z_n\).