The Carlson-type zero-density theorem for the Beurling zeta function

Abstract

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.