  Given a frequency $\lambda = (\lambda_n)$ and $\ell \ge 0$, we introduce the
scale of Banach spaces $H_{\infty,\ell}^{\lambda}[Re > 0]$ of holomorphic
functions $f$ on the open right half-plane $[Re > 0]$, which satisfy $(A)$ the
growth condition $|f(s)| = O((1 + |s|)^\ell)$, and $(B)$ have a Riesz germ,
i.e. on some open subset and for some $m \ge 0$ the function $f$ coincides with
the pointwise limit (as $x \to \infty$) of the so-called $(\lambda,m)$-Riesz
means $\sum_{\lambda_n < x} a_n e^{-\lambda_n s}\big(
1-\frac{\lambda_n}{x}\big)^m ,\,x >0$ of some $\lambda$-Dirichlet series $\sum
a_n e^{-\lambda_n s}$. Reformulated in our terminology, an important result of
M. Riesz shows that in this case the function $f$ for every $k >\ell$ is the
pointwise limit of the $(\lambda,k)$-Riesz means of $D$ on $[Re > 0]$.
  Our main contribution is an extension -- showing that 'after translation'
every bounded set in $H_{\infty,\ell}^{\lambda}[Re > 0]$ is uniformly
approximable by all its $(\lambda,k)$-Riesz means of order $k>\ell$. This
follows from an appropriate maximal theorem, which in fact turns out to be at
the very heart of a seemingly interesting structure theory of the Banach spaces
$H_{\infty,\ell}^{\lambda}[Re > 0]$. One of the many consequences is that
$H_{\infty,\ell}^{\lambda}[Re > 0]$ basically consists of those holomorphic
functions on $[Re >0]$, which have a Riesz germ and are of finite uniform order
$\ell$ on $[Re >0]$.
  To establish all this and more, we need to reorganize (and to improve)
various aspects and keystones of the classical theory of Riesz summability of
general Dirichlet series as invented by Hardy and M. Riesz.
