The Carlson-type zero-density theorem for the Beurling ζ $\zeta$ 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 $m\in N$ (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.