Well-behaved Beurling primes and integers

In this paper, we study generalised prime systems for which both the prime and integer counting functions are asymptotically well-behaved, in the sense that they are approximately li(x) and ρ x, respectively (where ρ is a positive constant), with error terms of order O(x(θ 1)) and O(x(θ 2)) for some θ(1), θ(2) < 1. We show that it is impossible to have both θ(1) and θ(2) less than © 2004 Elsevier Inc. All rights reserved.