A Sigma-Based Detection Framework for Prime Constellations with Applications to Bounded Prime Gaps

Abstract

A family of prime constellation detectors D ( p , N ) expressed as rational powers of 2π, built from the sum-of-divisors function σ . The central object D ( p , N ) = ( 2 π ) p + N + 1 − [ σ ( p ) + σ ( p + 2 N ) ] / 2 equals 1 if and only if both p and p + 2 N are prime, and is strictly less than 1 otherwise. This construction generalizes naturally to k-tuples via a product formula. A zeta-regularized product identity ∏ D ( p ) = ( 2 π ) 3 8 is created and decomposed into an associated sum of three explicitly characterized sub-series whose regularized total equals −5/48, a value determined purely by ζ ( 0 ) and ζ ( − 1 ) . It is shown that the composite sub-series F ( 1 ) converges to a constant numerically close to log 10 2 . Zhang’s bounded-gap theorem is recast in this form, showing that the average sieve sum T ( x , M ) > 1 for M ≥ 6 , and extend the framework to k-prime clusters. The triple prime case D 3 ( p ) = 1 is proven to have exactly one solution ( p = 3) via a complete mod-3 obstruction, validating the framework in a case where the answer is known.