${\varOmega }$-result for the remainder term in Beurling’s prime number theorem for well-behaved integers
Volume 208 / 2023
Acta Arithmetica 208 (2023), 69-81
MSC: Primary 11N80; Secondary 11N56.
DOI: 10.4064/aa220516-20-3
Published online: 26 May 2023
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 of this, we prove how bounds on $\psi (x)-x$ lead to zero-free regions for the Beurling zeta function, generalizing a result of Pintz to the Beurling setting. This may be of independent interest.
