Zeros of functions in Hilbert spaces of Dirichlet series

The Dirichlet-Hardy space H-2 consists of those Dirichlet series Sigma(n)a(n)n(-s) for which Sigma(n) vertical bar a(n)vertical bar(2) < infinity. It is shown that the Blaschke condition in the half-plane Res > 1/2 is a necessary and sufficient condition for the existence of a nontrivial function in H-2 vanishing on a given bounded sequence. The proof implies in fact a stronger result: every function in the Hardy space H-2 of the half-plane Res > 1/2 can be interpolated by a function in H-2 on such a Blaschke sequence. Analogous results are proved for the Hilbert space D-alpha of Dirichlet series Sigma(n)a(n)n(-s) for which Sigma(n) vertical bar a(n)vertical bar(2) [d(n)](alpha) < infinity; here d(n) is the divisor function and a positive parameter. In this case, the zero sets are related locally to the zeros of functions in weighted Dirichlet spaces of the half-plane Res > 1/2. Partial results are then obtained for the zeros of functions in H-p (L-p analogues of H-2) for 2 < p < infinity , based on certain contractive embeddings of D-alpha in H-p.