  In a previous paper we proved a Carlson type density theorem for zeroes in
the critical strip for Beurling zeta functions satisfying Axiom A of
Knopfmacher. There we needed to invoke two additonal conditions, the
integrality of the norm (Condition B) and an "average Ramanujan Condition" for
the arithmetical function counting the number of different Beurling integers of
the same norm (Condition G).
  Here we implement a new approach of Pintz, using the classic zero detecting
sums coupled with Hal\'asz' method, but otherwise arguing in an elementary way
avoiding e.g. large sieve type inequalities. This way we give a new proof of a
Carlson type density estimate--with explicit constants--avoiding any use of the
two additional conditions, needed earlier.
  Therefore it is seen that the validity of a Carlson-type density estimate
does not depend on any extra assumption--neither on the functional equation,
present for the Selberg class, nor on growth estimates of coefficients say of
the "average Ramanujan type"-but is a general property, presenting itself
whenever the analytic continuation is guaranteed by Axiom A.
