  Given a frequency $\lambda=(\lambda_n)$, we study when almost all vertical
limits of a $\mathcal{H}_1$-Dirichlet series $\sum a_n e^{-\lambda_ns}$ are
Riesz-summable almost everywhere on the imaginary axis. Equivalently, this
means to investigate almost everywhere convergence of Fourier series of
$H_1$-functions on so-called $\lambda$-Dirichlet groups, and as our main
technical tool we need to invent a weak-type $(1, \infty)$ Hardy-Littlewood
maximal operator for such groups. Applications are given to $H_1$-functions on
the infinite dimensional torus $\mathbb{T}^\infty$, ordinary Dirichlet series
$\sum a_n n^{-s}$, as well as bounded and holomorphic functions on the open
right half plane, which are uniformly almost periodic on every vertical line.
