Restricted Goldbach Sums in Arithmetic Progressions II: Almost-All Theorems for General Moduli

Source abstract

For $q≥1$ and $gcd(a,q)=1$ we study the restricted weighted Goldbach sum $R_{a,q}(N)=\sum_{p_{1}+p_{2}=N,\; p_{1}\equiv a\,(q)} (logp_{1})(logp_{2})$. We show that an odd prime divisor of $q$ forces $R_{a,q}(N)=O_{q}(logN)$ on a set of even $N$ of positive density, so that a main term of size $≍N/φ(q)$ can hold on almost all $N$ only for $q=2^{k}$. For $q=2^{k}$ we state the mean-square estimate $\sum_{N\leq X} |R_{a,q}(N)-M_{a,q}(N)|^{2}{\ll }_{A,q}X^{3}(logX)^{-A}$ for every $A>0$, with $M_{a,q}(N)=2C_{2}\varphi (q)^{-1}\mathfrak{S}(N)N$, and derive from it that the exceptional set $\{N\leq X:R_{a,q}(N)=0\}$ has size ${\ll }_{A,q}X(logX)^{-A}$ and that $|R_{a,q}(N)-M_{a,q}(N)|\leq N(logN)^{-3}$ for all but $O_{A,q}(X(logX)^{-A})$ even $N≤X$. A ternary anchoring transfer and an unconditional quaternary existence statement via Helfgott’s theorem follow. Part II proves an unconditional limit law: for every fixed integer $h≠0$, the odd part ${\mathfrak{S}}_{0}(n)=\prod_{\mathrm{\ell }\mid n,\mathrm{\ell }\gt 2} (\mathrm{\ell }-1)/(\mathrm{\ell }-2)$ of the Hardy–Littlewood singular series, evaluated at the shifted primes $n=p+h$, converges in distribution to an explicit random Euler product $Y_{h}=\prod_{\mathrm{\ell }\gt 2,\mathrm{\ell }\nmid h} a_{\mathrm{\ell }}^{B_{\mathrm{\ell }}}$ with independent Bernoulli factors $\mathbb{P}(B_{\mathrm{\ell }}=1)=1/(\mathrm{\ell }-1)$; we establish convergence of all integer moments, an entire Mellin transform of order at most one, non-atomicity, unbounded support, a superpolynomial upper tail, and strict amplification relative to generic integers. Part III applies this law to the amplitude of the restricted main term along $N=p+h$ and, under a stated Hardy–Littlewood-type hypothesis, to the additive richness $R(p)$ of the prime-generation DAG. Part IV records, as computational findings with stated ranges, an inverse correlation $corr(R,H)≈-0.7935$ between richness and hub count over the first 5000 primes, of which conditioning on ${\mathfrak{S}}_{0}(p+1)$ explains about 8%. All numerical constants are enclosed by truncation of the relevant Euler products at P={10}^{7} with an explicit tail bound. The paper does not prove Goldbach’s conjecture or finiteness of the exceptional set.