The power series \(1+\frac z{\Gamma(1+i)}+\frac{z^2}{\Gamma (1+2i)}+\frac{z^3}{\Gamma(1+3i)}+\cdots\) has a natural boundary! (Q544900)

For each pair \(u\in\mathbb{C}^+:= \{z: \text{Re\,}z> 0\}\) and \(v\in\mathbb{R}^+:= \{x\in\mathbb{R}: x> 0\}\), the authors show that the non-lacunary power series \(\sum_{n\in\mathbb{Z}} \Gamma(u+ {2ivn\over\pi})\, z^n\) has \(\partial C_v\) as a natural boundary, where \(C_v:=\{z\in\mathbb{C}: e^{-|v|}< |z|< e^{|v|}\}\). The proof involves transforming this series into a lacunary Dirichlet series.