Zeros of functions in Hilbert spaces of Dirichlet series

Abstract

The Dirichlet–Hardy space \({\fancyscript{H}}^2\) consists of those Dirichlet series \(\sum _n a_n n^{-s}\) for which \(\sum _n |a_n|^2 1/2\) is a necessary and sufficient condition for the existence of a nontrivial function \(f\) in \({\fancyscript{H}}^2\) 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 \(\text{ Re} s>1/2\) can be interpolated by a function in \({\fancyscript{H}}^2\) on such a Blaschke sequence. Analogous results are proved for the Hilbert space \({\fancyscript{D}}_\alpha \) of Dirichlet series \(\sum _n a_n n^{-s}\) for which \(\sum _n |a_n|^2[d(n)]^\alpha 1/2\). Partial results are then obtained for the zeros of functions in \({\fancyscript{H}}^p\) (\(L^p\) analogues of \({\fancyscript{H}}^2\)) for \(2<p<\infty \), based on certain contractive embeddings of \({\fancyscript{D}}_\alpha \) in \({\fancyscript{H}}^p\).