New zero-density estimates for the Beurling $\zeta$ function

Abstract

In two previous papers the second author proved some Carlson type density theorems for zeroes in the critical strip for Beurling zeta functions satisfying Axiom A of Knopfmacher. In the first of these invoking two additonal conditions were needed, while in the second an explicit, fully general result was obtained. Subsequently, Frederik Broucke and Gregory Debruyne obtained, via a different method, a general Carlson type density theorem with an even better exponent, and recently Frederik Broucke improved this further, getting $N(\sigma,T) \le T^{a(1-\sigma)}$ with any $a>\dfrac{4}{1-\theta}$. Broucke employed a new mean value estimate of the Beurling zeta function, while he did not use the method of Hal\'asz and Montgomery. Here we elaborate a new approach of the first author, using the classical zero detecting sums coupled with a kernel function technique and Hal\'asz' method, but otherwise arguing in an elementary way avoiding e.g. mean value estimates for Dirichlet polynomials. We will make essential use of the additional assumptions that the Beurling system of integers consists of natural numbers, and that the system satisfies the Ramanujan condition, too. This way we give a new variant of the Carlson type density estimate with similar strength as Tur\'an's 1954 result for the Riemann $\zeta$ function, coming close even to the Density Hypothesis for $\sigma$ close to 1.