Mean values, order, and convergence in the Hardy space of Dirichlet series

Let ${\mathcal{H}}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. It follows from a theorem of Bohr that if $f$ in ${\mathcal{H}}^2$ has a bounded analytic continuation to the right half-plane, then its norm can be computed from the mean values $$\mathrm{\parallel }f{\mathrm{\parallel }}_{{\mathcal{H}}^2}^2=\underset{\sigma \to 0^{+}}{\lim }\underset{T\to \mathrm{\infty }}{\lim }\frac{1}{2T}{\int }_{-T}^T\mid f(\sigma +it){\mid }^{2}dt\mathrm{.}$$ We investigate to what extent the assumption that the analytic continuation be bounded can be relaxed, and obtain optimal results in two different directions. The first direction concerns the order of the analytic continuation, and yields as a corollary a version of the classical Landau–Schnee convergence theorem for ${\mathcal{H}}^2$. The second direction concerns the abscissa of convergence, and resolves a problem posed by McCarthy.