摘要：
据我们所知，本文给出了首个适用于所有充分大偶数的二元哥德巴赫猜想的条件性证明。所需条件为狄利克雷函数的广义黎曼猜想，以及两种附加谱假设中的任意一种。本文的方法与经典筛法和圆法有根本性不同。
对于每个待表示的偶数，我们以该偶数为模，利用奇狄利克雷特征标的正交性，精确提取平滑加权的哥德巴赫相关量。扣除精确对角项和由欧拉贝塔函数构造的模型后，得到一个算术余项。在广义黎曼猜想下，该余项在哈代空间上定义了一个紧自伴的希尔伯特—施密特算子。一个精确的二次型恒等式将加权哥德巴赫计数的误差与这一算子联系起来。由算子的谱构造的对称四元数函数，其零点位置决定了算子范数，从而控制计数误差。
在预先规定的无零区域假设下，我们证明了加权哥德巴赫计数的哈代—利特尔伍德渐近公式。在所有谱零点的实部均为二分之一这一更强假设下，我们进一步获得更精细的显式误差估计。任一谱假设与广义黎曼猜想结合，均可保证每个充分大的偶数具有达到经典预测数量级的素数对表示，且两个素数均位于该偶数的三分之一至三分之二之间。本文方法还可自然推广，用于研究其他加性数论问题。
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.
