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 Beurling (1937).
A Beurling generalized integer is a number that can be written as a product of Beurling generalized primes[definition needed]. 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.
Bateman, Paul T.; Diamond, Harold G. (1969), "Asymptotic distribution of Beurling's generalized prime numbers", in LeVeque, William Judson (ed.), Studies in Number Theory, M.A.A. studies in mathematics, vol. 6, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), pp. 152–210, ISBN978-0-13-541359-3, MR0242778
