Henry Helson meets other big shots — a brief survey

Abstract

A theorem of Henry Helson shows that for every ordinary Dirichlet series a n n −s with a square summable sequence (a n ) of coefficients, almost all vertical limits a n χ(n)n −s , where χ : N → 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.

Variants of a theorem of Helson on general Dirichlet series

“…Then for p = 2 statement (i) was proved by Hedenmalm and Saksman in [14], whereas Bayart in [1, Theorem 6] for every D ∈ H 1 proves the convergence of almost all vertical limits D ω on [Re > 0]. For Dirichlet series in H 2 Bayart deduces his theorem from the Menchoff-Rademacher theorem on almost everywhere convergence of orthonormal series (see also [8]), and extends it then to Dirichlet series H 1 by so-called hypercontractivity. In the general case statement (ii) for p = 2 is Helson's theorem 1.2 and under the more restrictive condition (BC) instead of (LC) and p = 1.…”