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

TL;DR

This paper investigates three distinct approaches—discrete sieving, spectral analysis, and probabilistic dynamics—to better understand and approximate the $n$-th prime number, bridging deterministic formulas and stochastic models.

Contribution

It introduces a novel discrete summation formula for $p_n$, refines a spectral model of prime emergence, and applies a growth process perspective inspired by Mertens' theorems.

Findings

Derived a new harmonic summation formula for $p_n$

Refined the spectral resonance model for prime distribution

Proposed a probabilistic growth model for prime spacing

Abstract

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 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.