On zero-density estimates for Beurling zeta functions

Abstract

We show the zero-density estimate \[ N(ζ_{\mathcal{P}}; α, T) \ll T^{\frac{4(1-α)}{3-2α-θ}}(\log T)^{9} \] for Beurling zeta functions $\hat{I}{\P }_{\mathcal{P}}$ attached to Beurling generalized number systems with integers distributed as $N_{\mathcal{P}}(x)=Ax+O(x^{\hat{I}}¸)$. 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 $\hat{I}{\P }_{\mathcal{P}}$, and discuss some optimality questions.