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

TL;DR

This paper develops a Riemann-von Mangoldt-type formula for Beurling generalized primes, providing explicit estimates on the associated zeta function and laying groundwork for future zero distribution analysis.

Contribution

It introduces a Riemann-von Mangoldt-type formula for Beurling primes and establishes explicit estimates on the Beurling zeta function under Axiom A, advancing understanding of its zeros.

Findings

Derived a formula for the summatory function of Beurling primes.

Established explicit bounds on the Beurling zeta function and its zeros.

Constructed a contour for integral transformations in the analysis.

Abstract

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.