Beurling integers with RH and large oscillation

Abstract

We construct a Beurling generalized number system satisfying the Riemann hypothesis and whose integer counting function displays extremal oscillation in the following sense. The prime counting function of this number system satisfies π(x)=Li(x)+O(x), while its integer counting function satisfies the oscillation estimate N(x)=ρx+Ω±(xexp⁡(−clog⁡xlog⁡log⁡x)) for some c>0, where ρ>0 is its asymptotic density. The construction is inspired by a classical example of H. Bohr for optimality of the convexity bound for Dirichlet series, and combines saddle-point analysis with the Diamond-Montgomery-Vorhauer probabilistic method via random prime number system approximations.