S. Revesz, Alfréd Rényi Institute of Math., Hungarian Academy of Sciences and IHP

Oscillation of the remainder term in the Beurling prime number formula

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 half-plane 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.