Zeros of functions in Hilbert spaces of Dirichlet series

The Dirichlet–Hardy space $\text{Ht}$ consists of those Dirichlet series $\sum_n a_n n^{-s}$forwhich$\sum_n |a_n|^2<\infty$. It is shown that the Blaschke condition in the half-plane $\mathrm{Re}s>1\mathrm{/}2$ is a necessary and sufficient condition for the existence of a nontrivial function $f$ in $\text{Ht}$ 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 $\mathrm{Re}s>1\mathrm{/}2$ can be interpolated by a function in $\text{Ht}$ on such a Blaschke sequence. Analogous results are proved for the Hilbert space $\text{Da}$ of Dirichlet series ${\sum }_n{a}_n{n}^{-s}$ for which $\sum_n |a_n|^2[d(n)]^\alpha <\infty$;here$d(n)$isthedivisorfunctionand$\alpha$ 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 $\operatorname{Re} s>1/2$\mathrm{.}Partialresultsarethenobtainedforthezerosoffunctionsin$\Hp$($L^p$analoguesof$\Ht$)for$2<p<\infty$, based on certain contractive embeddings of $\text{Da}$ in $\text{Hp}$.