Journal of Economic Intelligence and Mathematics

ABSTRACT

In this paper, which is entirely unconditional, we prove a sharpened almost-all theorem with fully explicit effective constants for the restricted weighted Goldbach sum

whose expected main term is $M_{a,q}\left(N\right)=C_{2}S\left(N\right)N/\phi \left(q\right)$, where $C_2=0.6601618\dots$ is the twin-prime constant and $S\left(N\right)$ is the binary singular series.

Our results are organised around four pillars. (I) We give a complete character-pair decomposition of the second moment of the error $E\left(N\right):=R_{a,q}\left(N\right)-M_{a,q}\left(N\right)$, extracting the exact diagonal constant $G/\left(2\phi \left(q\right)\right)$, where $G={\prod }_{p>2}^{}(1+{(p-1)}^{-2})\in [1.41320886,1.41320899]$ is the Gallagher–Goldston constant. (II) We establish a uniform minor-arc $L^4$ bound

by combining the complete Vaughan identity with the Bombieri–Vinogradov theorem in integral form, giving an explicit derivation of ${\kappa }_{explicit}=C_V^2{c}_{L^2}=4.004$ before applying a rigorous $10\%$ safety margin. (III) We derive the effective almost-all theorem

with the explicit constant $K:=2C(1,4)\le 38.82$, obtained from $C(1,4)\le 19.41$ via a Stechkin-type optimisation. (IV) We prove a Pintz-type exceptional-set bound on $\{N\le X:R_{a,q}\left(N\right)=0\}$.

Every statement in the main body carries the tag [PROVED]. No Generalised Riemann Hypothesis, no zero-density hypothesis, no ternary sum $W_{a,q}\left(n\right)$, no spectral input, and no Chen-type sieve are used anywhere.

Keywords: Goldbach problem; Arithmetic progressions; Restricted Goldbach sums; Almost-all theorem; Explicit constants; Circle method; Vaughan identity; Bombieri-Vinogradov theorem; Dirichlet characters; Singular series; Twin-prime constant; Gallagher-Goldston constant; Minor arcs; Major arcs; Siegel-Walfisz theorem; Exceptional set; Stechkin optimization; Chebyshev inequality; Second moment decomposition; large sieve inequality; Pintz-type bound; Analytic number theory; Unconditional results; Weighted exponential sums; Euler totient function