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arXiv · 2004.11501

Beurling integers with RH and large oscillation

Frederik Broucke·Gregory Debruyne·Jasson Vindas
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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 $π(x)= \operatorname*{Li} (x)+ O(\sqrt{x})$, while its integer counting function satisfies the oscillation estimate $N(x) = ρx + Ω_{\pm}\bigl(x\exp(-c\sqrt{\log x\log\log x})\bigr)$ 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.

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Frederik Broucke, Gregory Debruyne, Jasson Vindas. 2020-05-26. Beurling integers with RH and large oscillation. https://doi.org/10.1016/j.aim.2020.107240

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