Beurling zeta function

{{Short description|Riemann zeta function analogue replacing ordinary primes with Beurling generalized primes}}

In mathematics, a Beurling zeta function is an analogue of the Riemann zeta function where the ordinary primes are replaced by a set of Beurling generalized primes: any sequence of real numbers greater than 1 that tend to infinity. These were introduced by {{harvs|txt|last=Beurling|authorlink=Arne Beurling|year=1937}}.

A Beurling generalized integer is a number that can be written as a product of Beurling generalized primes{{Definition needed|date=June 2017}}. Beurling generalized the usual prime number theorem 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.