Dirichlet Series and Functional Analysis

Abstract

The study of Dirichlet series of the form $\sum\nolimits_{n = 1}^\infty {a_n n^{ - s} }$ has a long history beginning in the nineteenth century, and the interest was due mainly to the central role that such series play in analytic number theory. The general theory of Dirichlet series was developed by Hadamard, Landau, Hardy, Riesz, Schnee, and Bohr, to name a few. However, the main results were obtained before the central ideas of Functional Analysis became part of the toolbox of every analyst, and it would seem a good idea to insert this modern way of thinking into the study of Dirichlet series. Some effort has already been spent in this direction; we mention the papers by Helson [20, 21] and Kahane [22, 23]. However, the field did not seem to catch on. It is hoped that this paper can act as a catalyst by pointing at a number of natural open problems, as well as some recent advances. Fairly recently, in [17], Hedenmalm, Lindqvist, and Seip considered a natural Hilbert space H ² of Dirichlet series and began a systematic study thereof. The elements of H ² are analytic functions on the half-plane

$$ \mathbb{C}_{\frac{1} {2}} = \left\{ {s \in \mathbb{C} : Re s > \frac{1} {2}} \right\} $$

of the form

$$ f(s) = \sum\limits_{n = 1}^{ + \infty } {a_n n^{ - s} } $$

(1.1)

(1.1) where the coefficients a₁,a₂,a₃, ⋯ are complex numbers subject to the norm boundedness condition

$$ \left\| f \right\|_{\mathcal{H}^2 } = \left( {\sum\limits_{n = 1}^{ + \infty } {\left| {a_n } \right|^2 } } \right)^{\frac{1} {2}} < + \infty . $$