Spaces of Dirichlet series with the complete Pick property

We consider reproducing kernel Hilbert spaces of Dirichlet series with kernels of the form k(s, u) = ∑ ann −s−ū, and characterize when such a space is a complete Pick space. We then discuss what it means for two reproducing kernel Hilbert spaces to be “the same”, and introduce a notion of weak isomorphism. Many of the spaces we consider turn out to be weakly isomorphic as reproducing kernel Hilbert spaces to the Drury-Arveson space H2 d in d variables, where d can be any number in {1, 2, . . . ,∞}, and in particular their multiplier algebras are unitarily equivalent to the multiplier algebra of H2 d . Thus, a family of multiplier algebras of Dirichlet series are exhibited with the property that every complete Pick algebra is a quotient of each member of this family. Finally, we determine precisely when such a space of Dirichlet series is weakly isomorphic as a reproducing kernel Hilbert space to H2 d and when its multiplier algebra is isometrically isomorphic to Mult(H2 d).