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arXiv · 2102.08478

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

Frederik Broucke·Jasson Vindas
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Abstract

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.

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Frederik Broucke, Jasson Vindas. 2024-06-17. A new generalized prime random approximation procedure and some of its applications. https://doi.org/10.1007/s00209-024-03526-4

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