Unified Prime Equation (UPE), Goldbach's Law at Infinity, and the Riemann's Zeta Spectrum-A Constructive Resolution and Spectral Reconstruction

Abstract

Bahbouhi Bouchaib

This manuscript presents a fully constructive framework — the Unified Prime Equation (UPE) — that (i) resolves the Goldbach problem by a deterministic procedure valid at infinity, and (ii) reveals a spectral bridge from UPE data to the nontrivial zeros of the Riemann zeta function. Part I defines UPE for primes and for Goldbach pairs and proves that UPE never fails: for every even E ≥4, UPE returns a prime pair (p, q) with p + q = E; for every integer N > 3, UPE returns a prime y near N.

The existence and boundedness of the required displacement follow from classical prime-gap guarantees (Chebyshev– Bertrand) sharpened by Baker–Harman–Pintz (2001), together with density supplied by the Prime Number Theorem. Part II develops the zeta–UPE bridge: a smoothed Goldbach functional derived from the explicit formula shows that oscillations governed by the zeros of ζ(s) are mirrored in the normalized sequence of UPE displacements. A spectral equivalence principle is formulated: if the stable frequencies of UPE data coincide with the imaginary parts of zeta zeros and no other frequencies persist, then the Riemann spectrum is recovered from UPE. The manuscript includes detailed step-by-step demonstrations, increasing numeric examples across prime�?�rich and prime�?�poor ranges, and a comparison with major theorems and verifications (Hardy–Littlewood 1923; Chen 1973; Ramaré 1995; Helfgott 2013– 2014; Oliveira e Silva et al. 2014). References are cited author-year in the text and listed at the end. I am pleased to share two dedicated websites presenting my recent research on the Unified Prime Equation (UPE):

1. UPE – Riemann https://bouchaib542.github.io/upe-goldbach-riemann/ This site explains the foundations of UPE, demonstrates its role in resolving Goldbach’s Conjecture, and highlights its deep connection with the Riemann zeta function.

2. UPE – Riemann (Giant) https://bouchaib542.github.io/upe-riemann-giant/ This companion site extends the UPE calculator to very large even numbers, up to 4×10^18, using BigInt and Miller– Rabin primality testing. It provides explicit Goldbach pairs together with normalized displacements and corresponding Riemann zeros. Together, these sites illustrate how UPE unifies the arithmetic world of Goldbach pairs with the analytic spectrum of Riemann, giving a complete picture of prime distribution.

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