Three expressions of the $n$-th prime number: discrete sieving, spectral analysis and probabilistic dynamics

The search for a closed-form expression of the $n$-th prime number, $p_n$, has long oscillated between the rigid determinism of analytic functions and the apparent randomness of local distributions. This paper explores three different approaches to $p_n$. The first one formalizes an analytical identity for $p_n$ based on a harmonic summation filtered by a Möbius-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.