  We consider reproducing kernel Hilbert spaces of Dirichlet series with
kernels of the form $k(s,u) = \sum a_n n^{-s-\bar u}$, 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
$H^2_d$ in $d$ variables, where $d$ can be any number in $\{1,2,\ldots,
\infty\}$, and in particular their multiplier algebras are unitarily equivalent
to the multiplier algebra of $H^2_d$. 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 to $H^2_d$
and when its multiplier algebra is isometrically isomorphic to $Mult(H^2_d)$.
