Given \(\beta \in (0,1)\), we show the existence of a Beurling generalized number system whose integer counting satisfies \(N(x) = ax + O\bigl (x\exp (-c\log ^{\beta } x)\bigr )\) for some \(a>0\) and \(c>0\), and whose prime counting function satisfies \(\pi (x) = {{\,\mathrm{Li}\,}}(x) + \varOmega \bigl (x\exp (-c'(\log x)^{\frac{\beta }{\beta +1}})\bigr )\) for some \(c'>0\). This is done by generalizing a construction of Diamond, Montgomery, and Vorhauer. This Beurling system serves as additional motivation for a conjecture of Bateman and Diamond from 1969, concerning the prime number theorem with Malliavin-type remainder.

给定\(\beta \in (0,1)\)，我们证明存在一个 Beurling 广义数系统，其整数计数满足\(N(x) = ax + O\bigl (x\exp (-c\log ^{\beta } x)\bigr )\)对于某些\(a>0\)和\(c>0\)，并且其素数计数函数满足\(\pi (x) = {{\,\mathrm {Li}\,}}(x) + \varOmega \bigl (x\exp (-c'(\log x)^{\frac{\beta }{\beta +1}})\bigr )\)为一些\(c'>0\)。这是通过概括 Diamond、Montgomery 和 Vorhauer 的构造来完成的。这个 Beurling 系统为 1969 年 Bateman 和 Diamond 的猜想提供了额外的动机，该猜想涉及具有 Malliavin 型余数的素数定理。