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An effective analytic recurrence for prime numbers

Benoit Cloitre

TL;DR

The paper converts the non-constructive Golomb–Keller prime recurrence, originally involving a limit as $s\to\infty$, into a constructive method by proving that a finite parameter $s_n\le p_n$ suffices when paired with a ceiling. It derives tight bounds on $s_n$ (notably $s_n\le 2p_n$ and $s_n\le p_n$ via Bertrand and Nagura), establishes $\liminf_{n} s_n/p_n=0$ and $0.3823\lesssim \limsup s_n/p_n\le 0.4332$, and identifies a threshold constant $c_0\approx0.5956$ governing fixed-coefficient regimes. The work further connects $s_n$ to prime constellations, provides empirical distributional evidence for $\sigma_n=s_n/p_n$ (Beta-like with mean around 0.29), and extends the constructive framework to Dirichlet $L$-functions, enabling residue predictions modulo small integers through sign information. Overall, the approach yields a practical, constructive recurrence for primes with deep links to prime gaps, constellations, and Hardy–Littlewood-type conjectures, and it opens avenues for arithmetic information extraction via generalized $L$-functions.

Abstract

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.

An effective analytic recurrence for prime numbers

TL;DR

The paper converts the non-constructive Golomb–Keller prime recurrence, originally involving a limit as , into a constructive method by proving that a finite parameter suffices when paired with a ceiling. It derives tight bounds on (notably and via Bertrand and Nagura), establishes and , and identifies a threshold constant governing fixed-coefficient regimes. The work further connects to prime constellations, provides empirical distributional evidence for (Beta-like with mean around 0.29), and extends the constructive framework to Dirichlet -functions, enabling residue predictions modulo small integers through sign information. Overall, the approach yields a practical, constructive recurrence for primes with deep links to prime gaps, constellations, and Hardy–Littlewood-type conjectures, and it opens avenues for arithmetic information extraction via generalized -functions.

Abstract

The Golomb--Keller formula expresses the next prime as a recurrence relation in terms of the first primes using the Riemann zeta function and an Euler product, but requires taking a limit as , rendering it non-constructive. We transform this asymptotic formula into an effective recurrence by proving that a finite parameter suffices when combined with the ceiling function, establishing a constructive method valid for all . The minimal integer parameter (OEIS A389650) reveals deep connections to prime constellations. We prove unconditionally, where . The limit superior satisfies , where is the supergolden ratio. The lower bound is conditional on the twin prime conjecture; the upper bound is unconditional. The constant 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 but also for predicting residues of modulo any integer with reduced precision requirements.

Paper Structure

This paper contains 25 sections, 16 theorems, 37 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The effective approach
  3. Main results
  4. Golomb's unifying framework
  5. From asymptotic to effective formula
  6. Analytical tools
  7. Dirichlet series machinery
  8. Convergence properties
  9. First effective bound
  10. Worked example: computing $p_7$ from $n=6$
  11. Sharp bound and asymptotic analysis
  12. Unconditional proof of $s_n \le p_n$
  13. Asymptotic behavior of $\sigma_n$
  14. Empirical distribution of $\sigma_n$
  15. Fixed coefficient formulas
  16. ...and 10 more sections

Key Result

Lemma 1

Let $s > 1$ and $m \geq 2$ be an integer.

Figures (3)

  • Figure 1: The ratio $\sigma_n = s_n/p_n$ for $n=1$ to $200$. The horizontal lines indicate the theoretical bounds for $C = \limsup \sigma_n$: $\log \psi \approx 0.382$ (conditional lower bound) and $0.4332$ (unconditional upper bound).
  • Figure 2: Histogram of $\sigma_n$ for $n=1$ to 200. The distribution is heavily skewed right. Vertical dashed lines indicate the theoretical bounds for $C$: $\log \psi \approx 0.382$ (green) and $0.4332$ (red).
  • Figure 3: Histogram of $\sigma_n$ ($n=1$ to 200) after removing 5 outliers identified by Tukey's fences. The core distribution is unimodal and moderately right-skewed, closely resembling a Beta(7.64, 18.62) distribution.

Theorems & Definitions (38)

  • Lemma 1: Integral bounds
  • proof
  • Definition 2
  • Lemma 3: Euler product
  • proof
  • Lemma 4
  • proof
  • Lemma 5: Monotonicity of $h$
  • proof
  • Proposition 6: Existence and characterization of $s_n^*$
  • ...and 28 more