论文标题
带有RH和大振荡的Beurling Integer
Beurling integers with RH and large oscillation
论文作者
论文摘要
我们构建了一个满足Riemann假设的概括性数字系统，其整数计数函数在以下意义上显示了极端振荡。此数字系统的素数功能满足$π（x）= \ peratatorName*{li}（x） + o（\ sqrt {x}）$，而其整数计数函数满足振荡估计估计估计估计值x}）\ bigr）$对于某些$ c> 0 $，其中$ρ> 0 $是其渐近密度。该结构的灵感来自H. bohr的经典示例，以实现DIRICHLET系列的凸度结合的最佳性，并将鞍点分析与钻石 - 蒙蒙特·沃尔霍尔（Diamond-Montgomery-Vorhauer-Vorhauer）通过随机质数系统近似值结合在一起。
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.
