An effective analytic recurrence for prime numbers

The Golomb–Keller formula expresses the next prime $p_{n+1}$ as a recurrence relation in terms of the first $n$ primes $p_1,\dots ,p_n$ using the Riemann zeta function and an Euler product, but requires taking a limit as $s\to \mathrm{\infty }$, rendering it non-constructive. We transform this asymptotic formula into an effective recurrence by proving that a finite parameter $s\le p_n$ suffices when combined with the ceiling function, establishing a constructive method valid for all $n\ge 1$. The minimal integer parameter $s_n$ (OEIS A389650) reveals deep connections to prime constellations. We prove ${\mathrm{lim\hspace{0.17em}inf}}_{n\to \mathrm{\infty }}{\sigma }_n=0$ unconditionally, where ${\sigma }_n=s_n\mathrm{/}{p}_n$. The limit superior $C=\mathrm{lim\hspace{0.17em}sup}{\sigma }_n$ satisfies $\log \psi \lesssim C\le 0.4332$, 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.