Zeros of functions in Hilbert spaces of Dirichlet series

الأصل · EN

The Dirichlet--Hardy space consists of those Dirichlet series ∑ₙ aₙ n⁻ˢ for which ∑ₙ |aₙ|²<∞. It is shown that the Blaschke condition in the half-plane Re s>1/2 is a necessary and sufficient condition for the existence of a nontrivial function f in vanishing on a given bounded sequence. The proof implies in fact a stronger result: every function in the Hardy space H² of the half-plane Re s>1/2 can be interpolated by a function in on such a Blaschke sequence. Analogous results are proved for the Hilbert space of Dirichlet series ∑ₙ aₙ n⁻ˢ for which ∑ₙ |aₙ|²[d(n)]α<∞; here d(n) is the divisor function and α 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 Re s>1/2. Partial results are then obtained for the zeros of functions in (Lᵖ analogues of) for 2<p<∞, based on certain contractive embeddings of in.