On zero-density estimates for Beurling zeta functions

TL;DR

This paper establishes a zero-density estimate for Beurling zeta functions linked to generalized number systems with specific distribution properties, advancing understanding of their zeros and potential implications for number theory.

Contribution

It provides a new zero-density estimate for Beurling zeta functions and extends the analysis to broader classes of Dirichlet series, including discussions on potential improvements.

Findings

Derived explicit zero-density bounds for Beurling zeta functions.

Extended estimates to broader classes of Dirichlet series.

Discussed potential improvements under additional bounds.

Abstract

We show the zero-density estimate \[ N(\zeta_{\mathcal{P}}; \alpha, T) \ll T^{\frac{4(1-\alpha)}{3-2\alpha-\theta}}(\log T)^{9} \] for Beurling zeta functions $\zeta_{\mathcal{P}}$ attached to Beurling generalized number systems with integers distributed as $N_{\mathcal{P}}(x) = Ax + O(x^{\theta})$. We also show a similar zero-density estimate for a broader class of general Dirichlet series, consider improvements conditional on finer pointwise or $L^{2k}$-bounds of $\zeta_{\mathcal{P}}$, and discuss some optimality questions.