Some examples of well-behaved Beurling number systems
Frederik Broucke, Gregory Debruyne, Szilárd Révész
TL;DR
The paper investigates the existence of well-behaved Beurling number systems, seeking power savings in both prime and integer counting functions. It introduces a two-stage construction: first build extended-sense template systems by prescribing zeros and poles of the Beurling zeta, then discretize to obtain actual Beurling primes via a discretization theorem. The authors prove that for every $\alpha\in[0,1)$ and $\beta\in[1/2,1)$ there exist $[\alpha,\beta]$-systems, and, under the Riemann hypothesis, construct families with $\beta<1/2$. They treat the challenging case $\alpha<\beta$ using a left-leaning growth construction, yielding $[\alpha,\beta]$-systems in new regions. The work also outlines methods to perturb classical prime–integer templates and discusses implications for algebraic number fields and the extended RH, contributing a broad landscape of Beurling systems with controlled error terms.
Abstract
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 $β<1/2$.