A Prime Gap Functional and its Spectral Link to the Zeros of the Riemann Zeta Function: A Structural Reduction of the Riemann Hypothesis

Francis Henry Ortiz Hidalgo · (2026) · DOI: 10.14293/pr2199.003480.v1

Resumen

We introduce a rigorous analytic framework linking the discrete distribution of consecutive prime gaps to the zeros of the Riemann zeta function ζ(s). The core object is a multiplicative functional K_i(s), defined on each consecutive prime pair (p_i, p_{i+1}), which encodes the local asymmetry of prime gaps. We establish a closed algebraic form for K_i(s) and demonstrate a fundamental factorization identity. By explicitly inserting Euler factors, the infinite product P(s) = ∏_{i=1}^{∞} K_i(s) is shown to satisfy the exact relation P(s) = ( ∏_{i=1}^{∞} C_i(s) ) · ζ(s)^2, where the product ∏ C_i(s) converges to a non-zero meromorphic function on the critical strip 0 < Re(s) < 1. Consequently, we prove the unconditional equivalence: P(1/2 + iγ) = 0 ⇔ ζ(1/2 + iγ) = 0. Furthermore, we provide a universal angular parametrization of the non-trivial zeros, showing that the condition Re(ρ) = 1/2 is algebraically equivalent to the relation tan(α) − tan(β) = 1/2, which serves as a geometric signature of the critical line. While these results do not constitute a complete proof of the Riemann Hypothesis, they rigorously reduce the problem to a single, well-defined analytical step: demonstrating that ζ(ρ) = 0 ⇒ K_i(Re(ρ)) = 0. We delineate this remaining gap and outline concrete strategies for its resolution, thereby providing a verifiable and structurally sound pathway for future investigation.

Instituciones: Universidad Nacional Santiago Antúnez de Mayolo