  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 $\pi(x) = \mathrm{Li}(x) +
\Omega\bigl(x\exp(-c'(\log x)^{\frac{\beta}{\beta+1}})\bigr)$ 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 PNT with Malliavin-type
remainder.
