Arne Beurling generalized the prime number theorem to the rather general situation when the role of primes are taken over by some arbitrary reals, and integers are simply the reals of the freely generated multiplicative subgroup of the primes given. If the "number of integers from 1 to $x$" function takes the form $N(x)=x + O(x^a)$, with $a<1$, then the corresponding Beurling zeta function has a meromorphic continuation to the halfplane to the right of $a$. Location of zeroes between the real part = 1 and real part $= a$ lines are then crucial to the oscillation of the prime number formula, as is well-known in the classical case. The lecture describes how these relations can be established even in the generality of Beurling prime distribution. In particular, we explain what (relatively mild) conditions can ensure that the most well-known classical zero density estimates carry over to the Beurling zeta function, too.