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

Jean-Christophe Pain

TL;DR

This paper investigates how to express the $n$-th prime using three distinct viewpoints that balance asymptotic density with discrete structure. It develops an exact discrete sieving formula via a Möbius-derived coprimality indicator, a phenomenological spectral resonance model based on von Mangoldt weighting and a Riemann-type oscillatory term, and a survival-dynamics/information-theoretic framework anchored in Mertens' theorem and Selberg sieve concepts. The first yields an exact identity for $p_{n+1}$ with computable sub-exponential complexity; the second offers a bounded spectral reconstruction whose primes act as points of constructive interference; the third models prime emergence as capacity-driven growth constrained by entropy, including capacity depletion and twin-prime considerations. Together, these approaches form a multi-scale framework that connects analytic number theory, sieve theory, and information theory, offering a richer picture of prime distribution beyond traditional asymptotics and suggesting new directions for understanding discrete arithmetic realities.

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 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.

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

TL;DR

This paper investigates how to express the -th prime using three distinct viewpoints that balance asymptotic density with discrete structure. It develops an exact discrete sieving formula via a Möbius-derived coprimality indicator, a phenomenological spectral resonance model based on von Mangoldt weighting and a Riemann-type oscillatory term, and a survival-dynamics/information-theoretic framework anchored in Mertens' theorem and Selberg sieve concepts. The first yields an exact identity for with computable sub-exponential complexity; the second offers a bounded spectral reconstruction whose primes act as points of constructive interference; the third models prime emergence as capacity-driven growth constrained by entropy, including capacity depletion and twin-prime considerations. Together, these approaches form a multi-scale framework that connects analytic number theory, sieve theory, and information theory, offering a richer picture of prime distribution beyond traditional asymptotics and suggesting new directions for understanding discrete arithmetic realities.

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 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.
Paper Structure (15 sections, 1 theorem, 43 equations)

This paper contains 15 sections, 1 theorem, 43 equations.

Table of Contents

  1. Introduction: the asymptotic baseline
  2. First formula: discrete sieving via Möbius-derived coprimality indicators
  3. Second formula: spectral perspective and harmonic resonance
  4. Third formula: "survival dynamics" and information entropy
  5. Mertens' theorem as a constraint
  6. Information entropy and the prime sieve
  7. Selberg's sieve and dynamics optimization
  8. The quadratic filtering, information entropy and the extremal principle
  9. The sieve capacity identity and structural density
  10. Capacity depletion and the twin prime conjecture
  11. Numerical discussion and implementation limits
  12. The precision bottleneck and certification
  13. Spectral resolution and Gibbs phenomenon in the second formula
  14. Asymptotic drift of survival dynamics in the third formula
  15. Conclusion

Key Result

Theorem 1

Let $(p_n)_{n\ge1}$ denote the sequence of prime numbers and $P_n = \prod_{i=1}^n p_i$ the $n$-th primorial. Let $\chi_n(m)$ be the arithmetical filter defined by: where $\mu$ is the Möbius function Mobius1832. Then, the $(n+1)$-th prime is the unique integer $m > 1$ in the range $[1, 2p_n]$ such that $\chi_n(m)=1$, which can be expressed as: Furthermore, the following harmonic identity holds:

Theorems & Definitions (2)

  • Theorem 1
  • proof