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 ) = ρ x + Ω ± ( x exp ⁡ ( − c log ⁡ x log ⁡ 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.

中文翻译：

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具有 RH 和大振荡的 Beurling 整数

摘要 我们构造了一个满足黎曼假设的Beurling广义数系统，其整数计数函数在以下意义上表现出极值振荡。该数系的素数计数函数满足 π ( x ) = Li ( x ) + O ( x ) ，而其整数计数函数满足振荡估计 N ( x ) = ρ x + Ω ± ( x exp ⁡ ( − c log ⁡ x log ⁡ log ⁡ x ) ) 对于某些 c > 0 ，其中 ρ > 0 是它的渐近密度。该构造的灵感来自 H. Bohr 的经典示例，用于优化狄利克雷级数的凸边界，并通过随机素数系统近似将鞍点分析与 Diamond-Montgomery-Vorhauer 概率方法相结合。