On PNT equivalences for Beurling numbers

TL;DR

This paper investigates the conditions under which various asymptotic relations in Beurling generalized numbers are equivalent to the prime number theorem, establishing new equivalences under weaker assumptions.

Contribution

It demonstrates that certain classical prime number theorem relations remain equivalent in the Beurling setting under less restrictive conditions than previously known.

Findings

Established new equivalences between Beurling PNT relations

Reduced the hypotheses needed for these equivalences

Extended understanding of Beurling number theory relations

Abstract

In classical prime number theory several asymptotic relations are considered to be "equivalent" to the prime number theorem. In the setting of Beurling generalized numbers, this may no longer be the case. Under additional hypotheses on the generalized integer counting function, one can however still deduce various equivalences between the Beurling analogues of the classical PNT relations. We establish some of the equivalences under weaker conditions than were known so far.