Henry Helson meets other big shots -- A brief survey

Abstract

A theorem of Henry Helson shows that for every ordinary Dirichlet series $\sum a_n{n}^{-s}$ with a square summable sequence $(a_n)$ of coefficients, almost all vertical limits $\sum a_n\chi (n)n^{-s}$, where $\chi :\mathbb{N}\to \mathbb{T}$ is a completely multiplicative arithmetic function, converge on the right half-plane. We survey on recent improvements and extensions of this result within Hardy spaces of Dirichlet series -- relating it with some classical work of Bohr, Banach, Carleson-Hunt, Cesàro, Hardy-Littlewood, Hardy-Riesz, Menchoff-Rademacher, and Riemann.