  In this paper we work out a Riemann-von Mangoldt type formula for the
summatory function $\psi(x):=\sum_{g\in G, |g|\le x} \Lambda_{G}(g)$, where $G$
is an arithmetical semigroup (a Beurling generalized system of integers) and
$\Lambda_{G}$ is the corresponding von Mangoldt function attaining $\log|p|$
for $g=p^k$ with a prime element $p\in G$ and zero otherwise. On the way
towards this formula, we prove explicit estimates on the Beurling zeta function
$\zeta_{G}$, belonging to $G$, to the number of zeroes of $\zeta_G$ in various
regions, in particular within the critical strip where the analytic
continuation exists, and to the magnitude of the logarithmic derivative of
$\zeta_G$, under the sole additional assumption that Knopfmacher's Axiom A is
satisfied. We also construct a technically useful broken line contour to which
the technic of integral transformation can be well applied. The whole work
serves as a first step towards a further study of the distribution of zeros of
the Beurling zeta function, providing appropriate zero density and zero
clustering estimates, to be presented in the continuation of this paper.
