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.We study the connection between the asymptotic behavior in terms of the inequalities (1) for general sequences (w n ) n∈N of non-negative numbers, and local 1 arXiv:1011.3370v1 [math.CV]

We obtain new results on Fourier multipliers for Dirichlet-Hardy spaces. As a consequence, we establish a Littlewood-Paley type inequality which yields a simple proof that the Dirichlet monomials form a Schauder basis for p > 1. fin a ν z ν , where z ∈ T ∞ and we use multi-index notation. The central observation, essentially2000 Mathematics Subject Classification. Primary 30B50; Secondary 42B15, 42B30, 46B15.

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We denote by $\Hp$ the Hilbert space of ordinary Dirichlet series with square-summable coefficients. The main result is that a bounded sequence of points in the half-plane $\sigma >1/2$ is an interpolating sequence for $\Hp$ if and only if it is an interpolating sequence for the Hardy space $H^2$ of the same half-plane. Similar local results are obtained for Hilbert spaces of ordinary Dirichlet series that relate to Bergman and Dirichlet spaces of the half-plane $\sigma >1/2$

a b s t r a c tWe study the zero sets of random analytic functions generated by a sum of the cardinal sine functions which form an orthonormal basis for the Paley-Wiener space. As a model case, we consider real-valued Gaussian coefficients. It is shown that the asymptotic probability that there is no zero in a bounded interval decays exponentially as a function of the length.