Relatively convergent series and product (Q1901192)

The author proves two theorems about convergence. I. Let \(\sigma\geq 0\), \(0< \theta\leq {\pi\over 2}\), and let \(f\) be a function holomorphic inside the angle \(|\arg z|< \theta\) such that \(|f(z)|\leq Ce^{\sigma|z|}\). If \(\sigma< \pi\sin \theta\) and \(f(n)\to 0\) as the integers \(n\to \infty\), then for \(\beta\in \mathbb{R}\), \(|\beta|< \pi- \sigma/\sin \theta\), the series \[ \sum^\infty_1 (- 1)^n e^{in\beta} f(n) \] converges and the convergence is uniform with respect to \(\beta\) on any inner segment. II. Consider the products \[ \prod^\infty_1 (1+ z^n \lambda_n), \] where \(|z|= 1\) and \(1> \lambda_1\geq \lambda_2\geq\cdots\geq \lambda_n\to 0\), and denote by \(\pi_n\) the product of the first \(n\) terms. Suppose that \(\sum_n \lambda^q_n= \infty\) for all positive integers \(q\). Then on the circle \(|z|= 1\) there exists an everywhere dense subset \(E\) of type \(G_\delta\) such that if \(z\in E\) and \(t\geq 0\), then \(|\pi_{n_k}|\to t\) for some subsequence \(n_k\). In particular the infinite product diverges for all \(z\) in \(E\).