Note on a conjecture of Bateman and Diamond concerning the abstract PNT with Malliavin-type remainder

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 $c>0$, prime function $\pi(x) \mathrm{Li}(x) \Omega\bigl(x\exp(-c'(\log x)^{\frac{\beta}{\beta+1}})\bigr)$ $c'>0$. This is done by generalizing construction Diamond, Montgomery, Vorhauer. serves as additional motivation conjecture Bateman Diamond from 1969, concerning PNT with Malliavin-type remainder.