Restricted Goldbach Sums in Arithmetic Progressions, 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 $$R_{a,q}(N):=\sum_{\substack{p_1+p_2=N\\p_1\equiv a\pmod q}}(\log p_1)(\log p_2).$$ We give a fully rigorous, self-contained treatment of the restricted binary, ternary and quaternary Goldbach problems in this setting. We first prove an elementary but decisive local obstruction: if an odd prime $\ell$ divides $q$, then $R_{a,q}(N)$ collapses to $O_q(\log N)$ 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 $$M_{a,q}(N)=\frac{C_2}{\varphi(q)}S(N)N,$$ Within this scope we prove a qualitative almost-all theorem, with a complete major/minor-arc derivation, showing the exceptional set has density zero. We record a conditional impossibility theorem showing that no $X$-independent threshold can upgrade this to an effective almost-all statement once a matching second-moment lower bound is granted, and we isolate, as honestly labelled structural cautions rather than theorems, four classical routes that fail to upgrade the almost-all theorem to unconditional finiteness. We add 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 $$S_0(n):=\prod_{\ell\nmid n}\frac{\ell-1}{\ell-2}$$ of the Hardy--Littlewood singular series, evaluated along shifted primes $n=p+h$ for fixed $h\neq 0$, converges in distribution to an explicit random Euler product $$Y_h=\prod_{\ell>2,\ell\nmid h}\left(\frac{\ell-1}{\ell-2}\right)B_\ell$$ with independent Bernoulli local factors $P(B_\ell=1)=1/(\ell-1)$; 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 $S(N)=S_0(N)$ 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 $N=p+h$, a connection neither source study states. Throughout, every asserted theorem is unconditional and every numerical constant is independently certified via partial Euler products with explicit tail bounds; statements retracted at an earlier stage of this programme after failing independent verification are recorded only as open problems.