  We show that the asymptotic behavior of the partial sums of a sequence of
positive numbers determine the local behavior of the Hilbert space of Dirichlet
series defined using these as weights. This extends results recently obtained
describing the local behavior of Dirichlet series with square summable
coefficients in terms of local integrability, boundary behavior, Carleson
measures and interpolating sequences. As these spaces can be identified with
functions spaces on the infinite-dimensional polydisk, this gives new results
on the Dirichlet and Bergman spaces on the infinite dimensional polydisk, as
well as the scale of Besov-Sobolev spaces containing the Drury-Arveson space on
the infinite dimensional unit ball. We use both techniques from the theory of
sampling in Paley-Wiener spaces, and classical results from analytic number
theory.
