A new generalized prime random approximation procedure and some of its applications

We present a new random approximation method that yields the existence of a discrete Beurling prime system \(\mathcal {P}=\{p_{1}, p_{2}, \cdots \}\) 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 Diamond et al. (Math Ann 334:1–36, 2006), and refined by Zhang (Math Ann 337:671–704, 2007). We obtain several applications. Our new method is applied to a question posed by 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 Hilberdink and Neamah (Int J Number Theory 16(05):1005–1011, 2020), and to improve the main result from (Adv Math 370:Article 107240, 2020), where a Beurling prime system with regular primes but extremely irregular integers was constructed.