$\varOmega$-result for the remainder term in Beurling’s prime number theorem for well-behaved integers

Abstract

We obtain a new $\varOmega $-result for the remainder term $\psi (x)-x$ of a Beurling prime system for which the integers are very well-behaved in the sense that $N(x)=ax + \mathrm O(x^\beta )$ for some $a \gt 0$ and $\beta \lt 1/2$. As part