An effective analytic recurrence for prime numbers

2025/07/23 by Benoit Cloitre, Cloitre, Benoit · 3 voices
Mathematics · #11B37 #11M06 #11Y55 #FOS: Mathematics #History and Overview (math.HO) #Number Theory (math.NT) #Primary 11A41 #Secondary 11N05 #math.HO #math.NT

paper · pdf · doi:10.48550/arxiv.2508.02690

arxiv published 2025/07/23 · arxiv updated 2025/10/12

Abstract

The Golomb--Keller formula expresses the next prime p_n+1 as a recurrence relation in terms of the first n primes p₁, …, p_n using the Riemann zeta function and an Euler product, but requires taking a limit as s → ∞, rendering it non-constructive. We transform this asymptotic formula into an effective recurrence by proving that a finite parameter s ≤ p_n suffices when combined with the ceiling function, establishing a constructive method valid for all n ≥ 1. The minimal integer parameter s_n (OEIS A389650) reveals deep connections to prime constellations. We prove \liminf_n→∞ σ_n = 0 unconditionally, where σ_n = s_n/p_n. The limit superior C = \limsup σ_n satisfies log ψ\lesssim C ≤ 0.4332, where ψ≈ 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.

Citations

• Convergence and generalization of a recursion equation for primes
• Estimates of Some Functions Over Primes without R.H.
• A recursion equation for prime numbers
• The Difference Between Consecutive Primes, II
• Approximate formulas for some functions of prime numbers

Discussions

• #arXiv An effective analytic recurrence for prime numbers: from asymptotics to explicit bounds arxiv.org/abs/2508.02690 A survey and a new contribution to the study of recurrence formulas for prime nu [bsky, 3 points, 0 comments]
• #arXiv An effective analytic recurrence for prime numbers: from asymptotics to explicit bounds https://arxiv.org/abs/2508.02690 A survey and a new contribution to the study of recurrence formulas for [bsky, 0 points, 0 comments]
• # arXiv An effective analytic recurrence for prime numbers: from asymptotics to explicit bounds https:// arxiv.org/abs/2508.02690 A survey and a new contribution to the study of recurrence formulas fo [mastodon, 0 points, 0 comments]

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