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

Volume 208 / 2023

Titus W. Hilberdink, Laima Kaziulytė Acta Arithmetica 208 (2023), 69-81 MSC: Primary 11N80; Secondary 11N56. DOI: 10.4064/aa220516-20-3 Published online: 26 May 2023


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.


  • Titus W. HilberdinkDepartment of Mathematics
    University of Reading
    Reading RG6 6AX, UK
  • Laima Kaziulytė 

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