Bohr's strips for Dirichlet series in Banach spaces (Q548501)

The Bohr-Bohnenblust-Hille Theorem states that the width of the largest possible strip in \(\mathbb{C}\) on which any given ordinary Dirichlet series \(\sum_{n\geq 1}a_nn^{-s}\) (with variable \(s\in\mathbb{C}\) and coefficients \(a_n\in\mathbb{C}\)) converges uniformly but not absolutely is equal to \(\frac{1}{2}.\) Bohr-Bohnenblust-Hille's ideas led to improvements and to new nice results, in particular to Dirichlet polynomials or Dirichlet series with coefficients in Banach spaces. The paper gives a survey of various aspects of these developments as well as several open problems.