Well-behaved Beurling number systems

A Beurling number system generalizes the multiplicative structure of the classical primes and integers. It consists of a non-decreasing unbounded sequence of real numbers $\{p_j{\}}_{j=1}^{\mathrm{\infty }}$ with $p_1>1$, called the generalized primes, and the sequence of generalized integers $\{n_k{\}}_{k=0}^{\mathrm{\infty }}$ which consists of the number 1 and all possible products of (powers of) the $p_j$. With such a system, one associates counting functions $\pi (x)$ and $N(x)$, counting the number of generalized primes and integers, respectively, below $x$. The primes satisfy the PNT if $\pi (x)\sim x/\log x$, and the integers have a density if $N(x)\sim \rho x$ for some positive $\rho$. If in these relations one has an error term of the form $O(x^a)$ for some $a<1$, one calls the primes or integers well-behaved.