  We present a new random approximation method that yields the existence of a
discrete Beurling prime system $\mathcal{P}=\{p_{1}, p_{2}, \dotso\}$ which is
very close in a certain precise sense to a given non-decreasing,
right-continuous, nonnegative, and unbounded function $F$. This discretization
procedure improves an earlier discrete random approximation method due to H.
Diamond, H. Montgomery, and U. Vorhauer [Math. Ann. 334 (2006), 1-36], and
refined by W.-B. Zhang [Math. Ann. 337 (2007), 671-704].
  We obtain several applications. Our new method is applied to a question posed
by M. Balazard concerning Dirichlet series with a unique zero in their half
plane of convergence, to construct examples of very well-behaved generalized
number systems that solve a recent open question raised by T. Hilberdink and A.
Neamah in [Int. J. Number Theory 16 05 (2020), 1005-1011], and to improve the
main result from [Adv. Math. 370 (2020), Article 107240], where a Beurling
prime system with regular primes but extremely irregular integers was
constructed.
