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 " function takes the form
, with , then the corresponding Beurling zeta
function has a meromorphic continuation to the halfplane to the
right of . Location of zeroes between the real part = 1 and
real part 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.