On conditional irrationality measures for values of the digamma function (Q868910)

For \(\theta\in\mathbb{R}\), let \(\beta=\beta(\theta)\in\mathbb{R}\) be the least number with the property that for any \(\varepsilon> 0\) there is \(q_0(\varepsilon> 0\) such that \(|\theta- p/q|>1/(\beta+ \varepsilon)^q\) for all \(p,q\in\mathbb{Z}\) with \(q\geq q_0(\varepsilon)\), and call it the irrationality base of \(\theta\). The authors give several irrationality criteria. One of them is as follows: Denote \(R_n= A_n\theta- B_n\) \((n= 1,2,\dots)\) for some \(A_n,B_n\in\mathbb{Z}\). Let \(\limsup_{n\to\infty}\log|A_n|/n\leq \sigma\), \(\sigma\geq 0\), and \(\lim_{n\to\infty}|nRn|=\tau\), \(\tau\geq 0\). (If \(\tau= 0\), suppose also that \(R_n\neq 0\) for all \(n\geq n_0\)) Then \(\theta\) is irrational and \(\beta(\theta)\leq e^{\sigma r}\). By using these criteria the authors obtain conditional irrationality measures for values at rational numbers of the digamma function \({\Gamma'\over\Gamma}(z)\).