Restricted Goldbach Sums in Arithmetic Progressions: An Effective Almost-All Theorem, Local Obstructions, and a Limit Law for the Singular-Series Bias on Shifted Primes

Source abstract

For $q \geq 1$ and $gcd(a,q) = 1$ consider the restricted weighted Goldbach sum $R_{a,q}(N): = \sum_{\substack{p_{1} + p_{2} = N \ p_{1} \equiv a\,(q)}}^{}(\log p_{1})(logp_{2})$. We give a fully rigorous, self-contained treatment of the restricted binary, ternary, and quaternary Goldbach problem in this setting. We first prove an elementary but decisive local obstruction: if an odd prime $\mathcal{l}$ divides $q$, then $R_{a,q}(N)$ collapses to $O_{q}(logN)$ on a positive-density set of even $N$, so no main term of size $\asymp N/\varphi(q)$ can hold uniformly; for $q = 2^{k}$ this obstruction is absent, and we identify the correct local main term $M_{a,q}(N) = 2C_{2}\,\mathfrak{S(}N)\, N/\varphi(q)$. Within this scope we prove, via a fully self-contained derivation of an explicit pointwise minor-arc bound (Vaughan's identity with the balanced parameters $U = V = X^{2/5}$), an exact master second-moment bound with leading constant $G/(2\varphi(q))$, and an effective almost-all theorem whose threshold is an explicit, necessarily $X$-growing function $C_{0}(A,q)(logX)^{(5 + A)/2}$; we prove separately that no threshold constant independent of $X$ can replace it once a matching second-moment lower bound is granted. We give a fully unconditional restricted Chen-type theorem obtained from the Selberg–Chen sieve and the classical Bombieri–Vinogradov theorem, a ternary prime-anchoring transfer, and positivity of the restricted quaternary singular series through explicit local densities. As a companion study, we then prove that the variable factor $\mathfrak{S}_{0}(n): = \prod_{\mathcal{l} \mid n,\mathcal{\, l} > 2}^{}\frac{\mathcal{l} - 1}{\mathcal{l} - 2}$ of the Hardy–Littlewood singular series, evaluated along shifted primes $n = p + h$, converges in distribution to an explicit random Euler product $Y_{h} = \prod_{\mathcal{l} > 2,\mathcal{\, l} \nmid h}^{}\left( \frac{\mathcal{l} - 1}{\mathcal{l} - 2} \right)^{B_{\mathcal{l}}}$ with independent Bernoulli local factors, $\mathbb{P(}B_{\mathcal{l}} = 1) = 1/(\mathcal{l} - 1)$; we identify its Mellin transform as an entire function of order one, prove convergence of every integral moment, and establish that the law is non-atomic, has unbounded support, and superpolynomially decaying tails, strictly amplified relative to generic integers. We then connect the two studies: since $\mathfrak{S(}N) = \mathfrak{S}_{0}(N)$ is exactly the amplitude of the restricted Goldbach main term $M_{a,q}(N)$, the limit law furnishes a rigorous probabilistic description of how that amplitude fluctuates as $N$ ranges over the shifted-prime sequence $N = p + h$. Throughout, every asserted theorem is unconditional and every numerical constant is independently certified via partial Euler products with explicit tail bounds; every genuinely conditional statement names its hypothesis in the statement itself, and statements that do not currently survive independent verification are recorded only as open problems.