Rational points of Euler's Gamma function (Q403037)

By following a method by \textit{D. Masser} [J. Number Theory 131, No. 11, 2037--2046 (2011; Zbl 1267.11091)] for Riemann's zeta function, the author obtains upper bounds for the number of rational values of Euler's gamma functions \(\Gamma(z)\) and \(1/\Gamma(z)\) for rationals \(z\) on an integral interval \([n-1,n]\). For example, we state the following result: there exists an absolute constant \(c>0\) such that for all integers \(n\geq 2\) and \(D\geq 3\), the number \(N\) of rationals \(z\) in \([n-1,n]\) with denominator at most \(n\) and such that \(\Gamma(z)\) is rational, verifies \(N\leq c\,n^4(\log n)^3\cdot(\log D)^2/\log\log D\). A similar result is shown for \(1/\Gamma(z)\).