Three expressions of the $n$-th prime number: discrete sieving, spectral analysis and probabilistic dynamics
Abstract
The search for a closed-form expression of the -th prime number, , has long oscillated between the rigid determinism of analytic functions and the apparent randomness of local distributions. This paper explores three different approaches to . The first one formalizes an analytical identity for based on a harmonic summation filtered by a M\"obius-derived coprimality indicator. Unlike Gandhi's 1971 identity, which employs a geometric density and logarithmic extraction, this formula operates through a discrete summation over the range defined by Bertrand's postulate. In the second one, we refine the ``harmonic resonance'' model, which posits that primes emerge as spectral nodes from von Mangoldt oscillations. Third, we adopt a ``survival dynamics'' approach, inspired by Mertens' theorems, treating prime spacing as an evolutionary growth process. By bridging these perspectives, we offer a comprehensive framework for understanding the transition from asymptotic trends to discrete arithmetic realities.
Keywords
Cite
@article{arxiv.2601.18816,
title = {Three expressions of the $n$-th prime number: discrete sieving, spectral analysis and probabilistic dynamics},
author = {Jean-Christophe Pain},
journal= {arXiv preprint arXiv:2601.18816},
year = {2026}
}