Some examples of well-behaved Beurling number systems

TL;DR

This paper explores the existence of well-behaved Beurling number systems with specific power savings in their counting functions, including constructions under the Riemann hypothesis.

Contribution

It demonstrates the existence of $[eta,eta]$-systems for various $eta$ and constructs special systems assuming the Riemann hypothesis.

Findings

Existence of $[eta,eta]$-systems for $eta o 1$

Construction of systems with $eta<1/2$ under Riemann hypothesis

Systems satisfy $ ext{max}\{ ext{alpha}, ext{beta}\} o 1/2$

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 $[\alpha,\beta]$-systems, where $\alpha$ and $\beta$ are connected to the optimal power saving in the prime and integer-counting functions. It is known that every $[\alpha,\beta]$-system satisfies $\max\{\alpha,\beta\}\ge1/2$. In this paper we show there are $[\alpha,\beta]$-systems for each $\alpha \in [0,1)$ and $\beta \in [1/2, 1)$. Assuming the Riemann hypothesis, we also construct certain families of $[\alpha,\beta]$-systems with $\beta<1/2$.