A Riemann-von Mangoldt-type formula for the distribution of Beurling primes

In this paper we work out a Riemann-von Mangoldt type formula for the summatory function $\psi (x):={\sum }_{g\in G,\mathrm{\mid }g\mathrm{\mid }\le x}{\mathrm{\Lambda }}_G(g)$, where $G$ is an arithmetical semigroup (a Beurling generalized system of integers) and ${\mathrm{\Lambda }}_G$ is the corresponding von Mangoldt function attaining $\log \mathrm{\mid }p\mathrm{\mid }$ 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.