Journal of Economic Intelligence and Mathematics

ABSTRACT

We establish the strongest unconditional consequence of combining two recent results of the author on the restricted weighted Goldbach sum

where $C_2=0.6601618\dots$ is the twin-prime constant and $S\left(N\right)$ is the binary singular series. Paper (unconditional) proves an almost-all bound: the number of even $N\le X$ for which $|R_{a,q}\left(N\right)-M_{a,q}\left(N\right)|>C(A,q)\hspace{0.17em}N{\left(logN\right)}^{-3}$ is at most $E_1(A,q)\hspace{0.17em}X{\left(logX\right)}^{-A}$, with the explicit constant $K:=2C(1,4)\le 38.02$. Paper(conditional on GRH) proves that $R_{a,q}\left(N\right)>0$ for every even $N\ge N_0\left(4\right)$ with $log{N}_0\left(4\right)=45.93$ (Normalisation B), i.e. $N_0\left(4\right)\approx {10}^{19.9}$; the certified worst-case bound is $log{N}_0\left(4\right)\le 46.1$.

We prove three results. (I) (Structural obstruction, unconditional) Combining the two papers to obtain $R_{a,4}\left(N\right)>0$ for every even $N\ge {10}^{19.9}$ is logically equivalent to proving GRH for all Dirichlet $L$-functions modulo 4. This barrier is structural, not a deficiency of the proofs. (II) (Main theorem, unconditional) For $q=4$, $a\in \{1,3\}$, any $A>0$, and all $X\ge max({10}^{20},e^{47})$,

where $E_1(A,4)$ is effectively computable; for $A=1$ the fraction of potential counterexamples among all even integers in $[{10}^{19.9},X]$ is at most $E_1(1,4)/logX\to 0$. (III) (Quantitative comparison with Pintz) The bound in (II) improves upon the Pintz unconditional bound $O\left(X^{0.72}\right)$  in relative density over $[{10}^{19.9},X]$: the exceptional set is $O_A\left(X{\left(logX\right)}^{-A}\right)$ for every $A>0$, which is smaller than any fixed power of $X$ in the relative sense.

This paper does not prove the Goldbach conjecture, does not prove GRH, and does not prove that every specific even $N\ge {10}^{19.9}$ is a Goldbach sum. The exceptional set is shown to have density zero but may be infinite. The result is the strongest unconditional statement extractable from the combination of and .

May 2026

Keywords: Goldbach problem; restricted Goldbach sum; arithmetic progressions; almost-all theorem; exceptional set; GRH threshold; Dirichlet $L$-functions; Montgomery–Vaughan theorem; Pintz bound; explicit constants; circle method.

MSC 2020: 11P32, 11N13, 11M26.