Composition operators and generalized primes

We study composition operators on the Hardy space ${\mathcal{H}}^2$ of Dirichlet series with square summable coefficients. Our main result is a necessary condition, in terms of a Nevanlinna-type counting function, for a certain class of composition operators to be compact on ${\mathcal{H}}^2$. To do that we extend our notions to a Hardy space ${\mathcal{H}}_{\mathrm{\Lambda }}^2$ of generalized Dirichlet series, induced in a natural way by a sequence of Beurling’s primes.