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 Ï(x)=Li(x)+O(x), while its integer counting function satisfies the oscillation estimate N(x)=Ïx+Ω±(xexpâĄ(âclogâĄxlogâĄlogâĄx)) for some c>0, where Ï>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.
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