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

In a previous paper, we proved a Carlson-type density theorem for zeroes in the critical strip for the Beurling zeta functions satisfying Axiom A of Knopfmacher. There we needed to invoke two additional 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 (Formula presented.) (Condition G). Here, we implement a new approach of Pintz using the classic zero-detecting sums coupled with Halász' method, but otherwise arguing in an elementary way avoiding, for example, large sieve-type inequalities or mean value estimates for Dirichlet polynomials. In 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 “average Ramanujan-type”—but is a general property presenting itself whenever the analytic continuation is guaranteed by Axiom A. © 2025 The Author(s). Journal of the London Mathematical Society is copyright © London Mathematical Society.