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Zeros of Functions in Bergman-Type Hilbert Spaces of Dirichlet Series

Ole Fredrik Brevig

Abstract


For a real number $\alpha$ the Hilbert space $\mathscr{D}_\alpha$ 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 $\mathscr{D}_\alpha$ 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 $\mathscr{H}^p$, for $1\leq p <2$.

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DOI: http://dx.doi.org/10.7146/math.scand.a-24745

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ISSN 0025-5521 (print) ISSN 1903-1807 (online)

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