Beurling zeta function

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 Beurling (1937).

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 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.

References

Bateman, Paul T.; Diamond, Harold G. (1969), "Asymptotic distribution of Beurling's generalized prime numbers", in LeVeque, William Judson, Studies in Number Theory, M.A.A. studies in mathematics, 6, Math. Assoc. Amer. (distributed by Prentice-Hall, Englewood Cliffs, N.J.), pp. 152–210, ISBN 978-0-13-541359-3, MR0242778
Beurling, Arne (1937), "Analyse de la loi asymptotique de la distribution des nombres premiers généralisés. I" (in French), Acta Mathematica (Springer Netherlands) 68: 255–291, doi:10.1007/BF02546666, ISSN 0001-5962, Zbl 0017.29604