  We study the distribution of zeros of zeta functions associated to Beurling
generalized prime number systems whose integers are distributed as $N(x) = Ax +
O(x^{\theta})$. We obtain in particular \[
  N(\alpha, T) \ll T^{\frac{c(1-\alpha)}{1-\theta}}\log^{9} T, \] for a
constant $c$ arbitrarily close to $4$, improving significantly the current
state of the art. We also investigate the consequences of the obtained
zero-density estimates on the PNT in short intervals. Our proofs crucially rely
on an extension of the classical mean-value theorem for Dirichlet polynomials
to generalized Dirichlet polynomials.
