Integral means and boundary limits of Dirichlet series

This paper deals with the boundary behaviour of functions in the Hardy spaces H(p) for ordinary Dirichlet series. The main result, answering a question of Hedenmalm, shows that the classical Carlson theorem on integral means does not extend to the imaginary axis for functions in H(infinity), that is, for the ordinary Dirichlet series in H(infinity) of the right half-plane. We discuss an important embedding problem for H(p), the solution of which is only known when p is an even integer. Viewing H(p) as Hardy spaces of the infinite-dimensional polydisc, we also present analogues of Fatou's theorem.