A Conditional Proof of the Goldbach Conjecture
Jiamin Li and Jianya Liu
Abstract
To the best of our knowledge, this paper presents the first conditional proof of the binary Goldbach conjecture for all sufficiently large even integers. The required assumptions are the Generalized Riemann Hypothesis for Dirichlet L-functions and either of two additional spectral hypotheses. Our approach differs fundamentally from classical sieve methods and the circle method.
For each even integer to be represented, we use the orthogonality of odd Dirichlet characters modulo that integer to extract exactly a smoothly weighted Goldbach correlation. Subtracting the exact diagonal contribution and a model constructed from Euler’s beta function yields an arithmetic remainder. Under the Generalized Riemann Hypothesis, this remainder defines a compact self-adjoint Hilbert–Schmidt operator on a Hardy space. An exact quadratic form identity relates the error in the weighted Goldbach count to this operator. The locations of the zeros of a symmetric quaternionic function constructed from the operator’s spectrum determine the operator norm, thereby controlling the counting error.
Under the hypothesis of a prescribed zero-free region, we establish the Hardy–Littlewood asymptotic formula for the weighted Goldbach count. Under the stronger hypothesis that all spectral zeros have real part one-half, we obtain sharper explicit error estimates. Either spectral hypothesis, combined with the Generalized Riemann Hypothesis, ensures that every sufficiently large even integer admits representations as a sum of two primes in numbers of the classically predicted order of magnitude, with both primes lying between one-third and two-thirds of that integer. Our method also extends naturally to other problems in additive number theory.
