In mathematics, a Beurling zeta function is an analogue of the Riemann zeta function where the ordinary primes are replaced by Beurling generalized primes: a sequence of real numbers greater than 1 that tend to infinity. These were introduced by .

A Beurling generalized integer is a number that can be written as a product of Beurling generalized primes. Beurling generalized the usual prime number theorem

Prime number theorem

In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....

to Beurling generalized primes. He showed that if the number N(x) of Beurling generalized integers less than x is of the form N(x) = Ax + O(x log^−γx) with γ > 3/2 then the number of Beurling generalized primes less than x is asymptotic to x/log x, just as for ordinary primes,
but if γ = 3/2 then this conclusion need not hold.