Restricted Goldbach Sums in Arithmetic Progressions II: Almost-All Theorems and a Shifted-Prime Limit Law
Year: 2026
Type: preprint
Abstract: For \( q≥1 \) and \( gcd(a,q)=1 \) we study the restricted weighted Goldbach sum \( R_{a,q}(N)=\sum_{p_{1}+p_{2}=N,\; p_{1}\equiv a\,(q)} (logp_{1})(logp_{2}) \), the sum being over primes. An elementary local obstruction shows that if an odd prime \( \ell \) divides \( q \), then on the set of even \( N≡a (mod \ell) \), of density \( 1/{\ell} \), the sum collapses to \( O_{q}(logN) \); hence a main term of size \( N/φ(q) \) can hold for almost all even \( N \) only for \( q=2^{k} \), and for general \( q \) the correct main term carries the admissibility condition \( gcd(N-a,q)=1 \). For \( q=2^{k} \) we prove, by the circle method with Vaughan’s minor-arc bound, the Siegel–Walfisz theorem on the major arcs and an exact evaluation of the singular-series coefficients, 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 \( A>0 \), an exceptional set of size \( O_{A,q}(X(logX)^{-A}) \), and an asymptotic formula with positive singular series for the restricted ternary sum, hence the representation of every large odd integer as a sum of three primes the first of which lies in a prescribed odd class modulo \( 2^{k} \). For general moduli we record the almost-all theorem for the corrected main term, conditional on a mean-value theorem of Tolev, and the local densities of the restricted ternary problem at every prime power dividing \( q \). We then prove that the odd part \( {\mathfrak{S}}_{0}(p+h) \) of the singular series along shifted primes converges in distribution to the explicit random Euler product \( Y_{h}=\prod_{\mathrm{\ell }\gt 2,\; \mathrm{\ell }\nmid h} {\left(\frac{\mathrm{\ell }-1}{\mathrm{\ell }-2}\right)}^{B_{\mathrm{\ell }}} \), with independent Bernoulli variables \( B_{\mathrm{\ell }} \) of mean \( 1/({\ell}-1) \), by a self-contained argument based on the Brun–Titchmarsh inequality. All integer moments converge to absolutely convergent Euler products, which are given in closed form as a function of the shift and enclosed in rigorous interval arithmetic for \( h=1 \) and \( h=3 \); the Mellin transform is entire of order at most one, with \( logM_{h}(\sigma )\leq \sigma \mathrm{loglog}\sigma +C_{0}\sigma \) . The law has no atoms, unbounded support and a doubly exponentially thin upper tail, \( \mathbb{P}(Y_{h}\gt y)\leq \exp(-\exp(c_{0}y)) \), which is sharp on a doubly logarithmic scale for every shift \( h \); by the Jessen–Wintner theorem the law is of pure type. Combining the two parts, we show that \( \varphi (q)R_{a,q}(p+h)/(2C_{2}(p+h)) \) converges in distribution to \( Y_{h} for q=2^{k} \) and odd \( h \), and, by the upper-bound sieve, that all its moments converge to those of \( Y_{h} \); in particular the mean number of restricted representations of \( p+1 \) is \( C_{2}A_{1}(1)\approx \) 1.1505 times that of a generic even integer of the same size. The relation to the additive richness of a prime-generation graph is established under a stated Hardy–Littlewood hypothesis. An inverse richness–hub correlation of about -0.79 over the first N primes, 500 ≤ \( N \) ≤ 20000, is recorded as a computational finding together with a size-controlled analysis: the finite window of the graph forces the hub count to decrease with the size of the prime, and once size and the singular-series factor are controlled jointly the residual correlation is of absolute value at most 0.20 and decreases with \( N \). Truncation-matched Kolmogorov distances between the empirical law of \( {\mathfrak{S}}_{0}(p+1) \) and the model are reported. All implied constants coming from the Siegel–Walfisz theorem are ineffective, and the paper proves neither Goldbach’s conjecture nor finiteness of any exceptional set.
Source: Preprints.org
Author: Ibar Federico Anderson
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7222920272