  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 $\beta<1/2$.
