An omega-result for Beurling generalized integers
Volume 212 / 2024
Acta Arithmetica 212 (2024), 359-371
MSC: Primary 11N80; Secondary 11M41
DOI: 10.4064/aa230324-20-11
Published online: 21 February 2024
Abstract
We consider Beurling number systems with very well-behaved primes, in the sense that $\psi (x) = x + O(x^{\alpha })$ for some $\alpha \lt 1/2$. We investigate how small the error term in the asymptotic formula for the integer-counting function $N(x)$ can be for such systems. In particular, we show that \[ N(x) - \rho x = \Omega (\sqrt{x}\,\mathrm e^{-(\log x)^{\beta }}) \] for any $\beta \gt 2/3$.
