  We study $H^p$ spaces of Dirichlet series, called $\mathcal{H}^p$, for the
range $0<p< \infty$. We begin by showing that two natural ways to define
$\mathcal{H}^p$ coincide. We then proceed to study some linear space properties
of $\mathcal{H}^p$. More specifically, we study linear functionals generated by
fractional primitives of the Riemann zeta function; our estimates rely on
certain Hardy--Littlewood inequalities and display an interesting phenomenon,
called contractive symmetry between $\mathcal{H}^p$ and $\mathcal{H}^{4/p}$,
contrasting the usual $L^p$ duality. We next deduce general coefficient
estimates, based on an interplay between the multiplicative structure of
$\mathcal{H}^p$ and certain new one variable bounds. Finally, we deduce general
estimates for the norm of the partial sum operator $\sum_{n=1}^\infty a_n
n^{-s}\mapsto \sum_{n=1}^N a_n n^{-s}$ on $\mathcal{H}^p$ with $0< p \le 1$,
supplementing a classical result of Helson for the range $1<p<\infty$. The
results for the coefficient estimates and for the partial sum operator exhibit
the traditional schism between the ranges $1\le p \le \infty$ and $0<p<1$.
