Zeros of functions in Hilbert spaces of Dirichlet series

Abstract

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