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 $\operatorname{Re} s>1/2$ 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 $\operatorname{Re} s>1/2$ 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 $\operatorname{Re} s>1/2$. 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 $\Hp$.