An effective analytic recurrence for prime numbers
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866909839101263872 |
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| author | Cloitre, Benoit |
| author_facet | Cloitre, Benoit |
| contents | The Golomb--Keller formula expresses the next prime $p_{n+1}$ as a recurrence relation in terms of the first $n$ primes $p_1, \ldots, p_n$ using the Riemann zeta function and an Euler product, but requires taking a limit as $s \to \infty$, rendering it non-constructive. We transform this asymptotic formula into an effective recurrence by proving that a finite parameter $s \leq p_n$ suffices when combined with the ceiling function, establishing a constructive method valid for all $n \geq 1$.
The minimal integer parameter $s_n$ (OEIS A389650) reveals deep connections to prime constellations. We prove $\liminf_{n\to\infty} σ_n = 0$ unconditionally, where $σ_n = s_n/p_n$. The limit superior $C = \limsup σ_n$ satisfies $\log ψ\lesssim C \leq 0.4332$, where $ψ\approx 1.46557$ is the supergolden ratio. The lower bound is conditional on the twin prime conjecture; the upper bound is unconditional. The constant $C$ relates to the densest admissible prime constellation, connecting to the Hardy--Littlewood conjectures.
The method extends to Dirichlet L-functions, yielding other effective formulas for calculating $p_{n+1}$ but also for predicting residues of $p_{n+1}$ modulo any integer with reduced precision requirements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_02690 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An effective analytic recurrence for prime numbers Cloitre, Benoit Number Theory History and Overview Primary 11A41, Secondary 11N05, 11B37, 11M06, 11Y55 The Golomb--Keller formula expresses the next prime $p_{n+1}$ as a recurrence relation in terms of the first $n$ primes $p_1, \ldots, p_n$ using the Riemann zeta function and an Euler product, but requires taking a limit as $s \to \infty$, rendering it non-constructive. We transform this asymptotic formula into an effective recurrence by proving that a finite parameter $s \leq p_n$ suffices when combined with the ceiling function, establishing a constructive method valid for all $n \geq 1$. The minimal integer parameter $s_n$ (OEIS A389650) reveals deep connections to prime constellations. We prove $\liminf_{n\to\infty} σ_n = 0$ unconditionally, where $σ_n = s_n/p_n$. The limit superior $C = \limsup σ_n$ satisfies $\log ψ\lesssim C \leq 0.4332$, where $ψ\approx 1.46557$ is the supergolden ratio. The lower bound is conditional on the twin prime conjecture; the upper bound is unconditional. The constant $C$ relates to the densest admissible prime constellation, connecting to the Hardy--Littlewood conjectures. The method extends to Dirichlet L-functions, yielding other effective formulas for calculating $p_{n+1}$ but also for predicting residues of $p_{n+1}$ modulo any integer with reduced precision requirements. |
| title | An effective analytic recurrence for prime numbers |
| topic | Number Theory History and Overview Primary 11A41, Secondary 11N05, 11B37, 11M06, 11Y55 |
| url | https://arxiv.org/abs/2508.02690 |