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 $\pi (x)=\mathrm{Li}(x)+O(\sqrt{x})$, while its integer counting function satisfies the oscillation estimate $N(x)=\rho x+{\Omega }_{\pm }(x\exp (-c\sqrt{\log x\log \log x}))$ for some $c>0$, where $\rho >0$ 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.

31 pages, 2 figures