A classical result of Fatou gives that every bounded holomorphic function on the disc has radial limits for almost every point in the torus, and the limit function belongs to the Hardy space H_\infty of the torus. This property is no longer true when we consider vector-valued functions. The Banach spaces X for which this property is satisfied are said to have the analytic Radon-Nikodym property (ARNP). Some important equivalent reformulations of ARNP are studied in this chapter. Among others, X has ARNP if and only if each X-valued H_p- function f on the disc has radial limits almost everywhere on the torus (and not only H_\infty-functions). Even more, in this case each such f has non-tangential limits within any Stolz region. The basic tools are subharmonic functions and certain maximal inequalities. Finally, it is shown that if X has the ARNP, then every L_p of functions taking values in X with a finite measure also has ARNP.

For each 1 ≤ p ≤ ∞, the Hardy space \mathcal{H}_p of Dirichlet series is defined as the image through the Bohr transform of the Hardy space of functions on the infinite-dimensional polytorus. The Dirichlet polynomials are dense in \mathcal{H}_p for every 1 ≤ p < ∞. For p=2 this coincides with the space of Dirichlet series whose coefficients are square-summable. A Dirichlet series with coefficients a_n belongs to\mathcal{H}_p if and only if the series with coefficients a_n/n^ε is in \mathcal{H}_p for every ε >0 and the norms are uniformly bounded. In this case, the series is the limit as ε tends to 0. As a technical tool to see this, vector-valued Dirichlet series (that is, series with coefficients in some Banach space) are introduced, and some basic definitions and properties (such as the convergence abscissas, Bohr-Cahen formulas) are given.