Some examples of well-behaved Beurling number systems
Frederik Broucke, Gregory Debruyne, Szilárd Gy. Révész · Repository of the Academy's Library (Library of the Hungarian Academy of Sciences) · 2023
We investigate the existence of well-behaved Beurling number systems, which are systems of Beurling generalized primes and integers which admit a power saving in the error term of both their prime and integer-counting function. Concretely, we search for so-called $[α,β]$-systems, where $α$ and $β$ are connected to the optimal power saving in the prime and integer-counting functions. It is known that every $[α,β]$-system satisfies $\max\{α,β\}\ge1/2$. In this paper we show there are $[α,β]$-systems for each $α\in [0,1)$ and $β\in [1/2, 1)$. Assuming the Riemann hypothesis, we also construct certain families of $[α,β]$-systems with $β<1/2$.