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

For a real number $\alpha$ the Hilbert spaces ${\mathcal{D}}_{\alpha }$ consists of those Dirichlet series ${\sum }_{n=1}^{\mathrm{\infty }}{a}_n\mathrm{/}{n}^s$ for which ${\sum }_{n=1}^{\mathrm{\infty }}\mathrm{\mid }{a}_n{\mathrm{\mid }}^2\mathrm{/}[d(n){]}^{\alpha }<\mathrm{\infty }$, 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 $1\le p<2$.