On zero-density estimates for Beurling zeta functions

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.