Some examples of well-behaved Beurling number systems

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 $\max \{\alpha ,\beta \}\ge 1\mathrm{/}2$. In this paper we show there are $[\alpha ,\beta ]$-systems for each $\alpha \in [0,1)$ and $\beta \in [1\mathrm{/}2,1)$. Assuming the Riemann hypothesis, we also construct certain families of $[\alpha ,\beta ]$-systems with $\beta <1\mathrm{/}2$.