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( x exp(- c )), and whose zeta function has infinitely many zeros on the curve σ =1− a /log t , 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.