New zero-density estimates for the Beurling zeta function
Tom 221 / 2025
Streszczenie
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 $a \gt \frac{4}{1-\theta }$. Broucke employed a new mean value estimate of the Beurling zeta function, without using the method of Halász 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 Halász’ method, but otherwise arguing in an elementary way avoiding e.g. mean value estimates for Dirichlet polynomials. We 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. This way we give a new variant of the Carlson type density estimate with strength similar to Turán’s 1954 result for the Riemann zeta function, coming close even to the Density Hypothesis for $\sigma $ close to $1$.
