An effective analytic recurrence for prime numbers

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 $\liminf_{n\to\infty} \sigma_n = 0$ unconditionally, where $\sigma_n = s_n/p_n$. The limit superior $C = \limsup \sigma_n$ satisfies $\log \psi \lesssim C \leq 0.4332$, where $\psi \approx 1.46557$ 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.