Arakelyan, N. U.; Martirosyan, V. A. Localization of singularities of power series on the boundary of the disk of convergence. (Russian. English summary) Zbl 0626.30002 Izv. Akad. Nauk Arm. SSR, Mat. 22, No. 1, 3-21 (1987). In the paper the authors investigate questions concerning the famous theorem of Fabry about singularities of power series of the form \[ (1)\quad \sum^{\infty}_{n=0}f_ nz^ n,\quad \overline{\lim}_{n\to \infty}| f_ n|^{1/n}=1. \] One of the main results is Theorem 1. Let two sequences \(\{n_ k\}_ 1^{\infty}\), \(n_ k\uparrow +\infty\), \(n_ k\in Z_+\), and \(\{\beta_ k\}_ 1^{\infty}\), \(\beta_ k\in {\mathbb{R}}\), so that \[ \overline{\lim}_{k\to \infty}| Re(f_{n_ k}e^{i\beta_ k})|^{1/n_ k}=1, \] where \(f_ n\) are the coefficients of the series (1). Then the series (1) has a singular point \(e^{i\omega}\), where \[ | \omega | \leq \pi \min \{\overline{\lim}_{\tau \to 1- 0}\overline{\lim}_{k\to \infty}(W_ k(\tau))/(| 1-\tau | n_ k),\quad \overline{\lim}_{\tau \to 1+0}\overline{\lim}_{k\to \infty}(W_ k(\tau))/(| 1-\tau | n_ k)\} \] and \(W_ k(\tau)\) is the number of sign changes in the sequence \(\{Re(f_ ne^{i\beta_ k})\}\) for \(n\in [\tau n_ k,n_ k]\), \(\tau\geq 0.\) From the main results the authors receive some corollaries which generalize Fabry’s theorems about gaps and ratios for power series. Reviewer: P.Z.Agranovich Cited in 3 ReviewsCited in 4 Documents MSC: 30B10 Power series (including lacunary series) in one complex variable Keywords:number of sign changes; Fabry’s theorems PDFBibTeX XMLCite \textit{N. U. Arakelyan} and \textit{V. A. Martirosyan}, Izv. Akad. Nauk Arm. SSR, Mat. 22, No. 1, 3--21 (1987; Zbl 0626.30002)