Some examples of well-behaved Beurling number systems

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

24 pages