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.