Szilárd Gy. Révész Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15, H-1053 Budapest, Hungary

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In this paper we work out a Riemann–von Mangoldt type formula for the summatory function ψx:= gG,gxΛGg, where G is an arithmetical semigroup (a Beurling generalized system of integers) and ΛG is the corresponding von Mangoldt function attaining logp for g =pk with a prime element pG and zero otherwise. On the way towards this formula, we prove explicit estimates on the Beurling zeta function ζG , belonging to G, to the number of zeroes of ζG in various regions, in particular within the critical strip where the analytic continuation exists, and to the magnitude of the logarithmic derivative of ζG, 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.

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