Zeros of functions in Bergman-type Hilbert spaces of Dirichlet Series

Abstract

For a real number $\alpha$ the Hilbert spaces ${\mathcal{D}}_{\alpha }$ consists of those Dirichlet series ${\sum }_{n=1}^{\infty }{a}_n/n^s$ for which ${\sum }_{n=1}^{\infty }|a_n{|}^2/[d(n){]}^{\alpha }<\infty$, where $d(n)$ denotes the number of divisors of $n$. We extend a theorem of Seip on the bounded zero sequences of functions in ${\mathcal{D}}_{\alpha }$ to the case $\alpha >0$. Generalizations to other weighted spaces of Dirichlet series are also discussed, as are partial results on the zeros of functions in the Hardy spaces of Dirichlet series ${\mathcal{H}}^p$, for $1\le p<2$.

arXiv commentMinor corrections to Sections 1 and 2. Section 3 reorganized