Zeros of sections of divergent power series (Q1916769)

Let \(\sum^\infty_{n = 0} c_n z^n\) be a power series with zero radius of convergence. Put \(s_n : = \sum^\infty_{k = 0} c_k z^k\) and let \(Z_n (\alpha,a)\) denote the set of all roots of the equation \(s_n (z/ \alpha) = a\). The authors prove the following four theorems: (1) For any \(\varepsilon > 0\) and \(a \in \mathbb{C}\) the set \(Z_n ({\root n \of {|c_n |}}, a)\) is contained in \(\{z : |c_0 - a |/(1 + |c_0 - a |) \leq |z |\leq 2 + \varepsilon\}\) for infinitely many \(n\). (2) If \(c_n \geq 0\) for all \(n \geq 0\) and \(c_{n - 1} c_{n + 1} > c^2_n\) for all \(n \geq 1\), then for any \(a \neq c_0\) the points from \(Z_n ({\root n \of {|c_n |}}, a)\) converge to the unit circle as \(n \to + \infty\). (3) If \(c_n = n!\) for all \(n \geq 0\), then for any \(a \neq 1\) the set \(Z (n/e, a)\) is contained in \(\{z : 1 - 3 (|1 - a |\sqrt n) < |z |< 1 + (1 + |a |) \sqrt {2/ (\pi n)}\}\) for all \(n \geq \max \{(16/9) e^4 |1 - a |^4\), \((3/ |1 - a |)^2\}\). (4) If \(c_n = n!\) for all \(n \geq 0\), then for any \(a \neq 1\) and \(0 \leq \theta_1 < \theta_2 \leq 2 \pi\) the quotient \((\# \{z \in Z (1,a) : \theta_1 \leq \arg z \leq \theta_2\})/n\) tends to \((\theta_2 - \theta_1)/(2 \pi)\) as \(n \to + \infty\).