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

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.

to appear in Mathematica Pannonica. arXiv admin note: text overlap with arXiv:2012.09045

References in corpus (1)

• Beurling integers with RH and large oscillation

Cited by in corpus (2)

• Distribution of Beurling primes and zeroes of the Beurling zeta function I. Distribution of the zeroes of the zeta function of Beurling
• On the Lindelöf hypothesis for general sequences