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We study the distribution of zeros of zeta functions associated to Beurlinggeneralized 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 aconstant $c$ arbitrarily close to $4$, improving significantly the currentstate of the art. We also investigate the consequences of the obtainedzero-density estimates on the PNT in short intervals. Our proofs crucially relyon an extension of the classical mean-value theorem for Dirichlet polynomialsto generalized Dirichlet polynomials.

On zero-density estimates and the PNT in short intervals for Beurling generalized numbers