Beurling primes with large oscillation

Abstract

A Beurling generalized number system is constructed having integer counting function N B (x) = κ x +O(x θ) with κ>0 and 1/2 <θ <1, whose prime counting function satisfies the oscillation estimate π B (x) =li(x) + Ω(xexp(-c )), and whose zeta function has infinitely many zeros on the curve σ=1−a/logt, t≥2, and no zero to the right of this curve, where a is chosen so that a>(4/e)(1−θ). The construction uses elements of classical analytic number theory and probability.