Bohr's radii and strips -- a microscopic and a macroscopic view (Q2902313)

This is a report on Bohr's radii and strips -- a microscopic and a macroscopic view. The aim of this article is to report on parts of this new development. The Bohr-Bohnenblust-Hille theorem states that the largest possible width \(S\) of the strip in the complex plane, on which a Dirichlet series \(\sum_n a_n 1/n^s\) converges uniformly but not absolutely, equals \(1/2\). In fact, \textit{H. Bohr} in 1913 [Gött. Nachr., 441--488 (1913; JFM 44.0306.01)] proved that \(S \leq 1/2\) and asked whether equality can be achieved. The general theory of Dirichlet series during this time was one of the most fashionable topics in analysis, and Bohr's so-called {absolute convergence problem} was very much in the focus. In this context Bohr himself discovered several deep connections of Dirichlet series and power series (holomorphic functions) in infinitely many variables, and as a sort of by-product he found his famous power series theorem. Finally, Bohnenblust and Hille in 1931 in a rather ingenious fashion answered the absolute convergence problem in the positive. In recent years many authors revisited the work of Bohr, Bohnenblust and Hille, improving this work but also extending it to more general settings, for example to Dirichlet series with coefficients in Banach spaces.