  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.
