Journal of Economic Intelligence and Mathematics

INTRODUCTION

The Gap Between Almost-All and All

The binary Goldbach conjecture asserts that every even integer $N\ge 4$ is the sum of two primes. In its analytic form, this is the statement $R\left(N\right)>0$ for all even $N\ge 4$, were

Montgomery and Vaughan [11] established the first quantitative almost-all theorem: the set of even $N\le X$ with $R\left(N\right)\ne C_{2}S\left(N\right)N+O\left(N{\left(logN\right)}^{-A}\right)$ has size $O_A\left(X{\left(logX\right)}^{-A}\right)$. Pintz improved the exceptional-set exponent to $O\left(X^{0.72}\right)$ unconditionally. Under GRH, the exceptional set is empty for $N\ge N_0$ with $N_0$ explicit.

The present paper is concerned with the restricted analogue, in which one prime is confined to an arithmetic progression modulo $q=4$, and with the strongest unconditional conclusion obtainable from the two companion papers 

The Two Input Theorems

We Recall Precisely The Two Results We Combine.

Theorem 1 (Paper 1 [1], unconditional). Fix $q=4$, $a\in \{1,3\}$, and $A>0$. There exist effectively computable constants $C(A,4)>0$ and $E_1(A,4)>0$ such that $\#\{N\le X\hspace{0.17em}:\hspace{0.17em}|R_{a,4}\left(N\right)-M_{a,4}\left(N\right)|>C(A,4)\hspace{0.17em}N{\left(logN\right)}^{-3}\}\hspace{0.17em}\le \hspace{0.17em}{E}_1(A,4)\hspace{0.17em}\frac{X}{{\left(logX\right)}^A}.$ For $A=1$ one has $C(1,4)\le 19.01$ and $K:=2C(1,4)\le 38.02$.

Theorem 2 (Paper 2 [2], conditional on GRH). Assume the Generalized Riemann Hypothesis for all Dirichlet $L$-functions modulo $4$. Then for every even $N\ge N_0\left(4\right)$, $R_{a,4}\left(N\right)=M_{a,4}\left(N\right)+O_{4,\epsilon }\left(N^{1/2+\epsilon }\right),\hspace{0.17em}\hspace{0.17em}{R}_{a,4}\left(N\right)\ge c_4\hspace{0.17em}\frac{N}{logN}>0,$ where the effective threshold satisfies $log{N}_0\left(4\right)=45.93$ (Normalisation B), with the certified worst-case bound $log{N}_0\left(4\right)\le 46.1$.

The Main Question And What This Paper Answers

A natural question is: can one combine Theorems 1 and 2 to obtain an unconditional result for all $N\ge {10}^{19.9}$?

We Give A Precise Three-Part Answer

No unconditional pointwise result ($R_{a,4}\left(N\right)>0$ for every $N\ge {10}^{19.9}$) follows from this combination; such a result would be equivalent to GRH (Section3).

Yes, an unconditional density result does follow: the number of potential counterexamples in $[{10}^{19.9},X]$ is $O_A\left(X{\left(logX\right)}^{-A}\right)$ (Section4).

The combination is genuinely informative, strictly stronger than either paper alone in the range $N\ge {10}^{19.9}$ (Section5).

What This Paper Does Not Claim

• It does not prove the binary Goldbach conjecture.

• It does not prove the restricted binary Goldbach conjecture for all $N\ge {10}^{19.9}$.

• It does not prove GRH.

• The exceptional set $\{N\ge {10}^{19.9}:R_{a,4}\left(N\right)=0\}$ is shown to have density zero, but may be infinite.

Notation And Fundamental Constants

Throughout, $p$, $p_1$, $p_2$ denote primes. We fix $q=4$ and $a\in \{1,3\}$ (the reduced residues modulo 4).

Definition 3 (Fundamental constants).

The error for a given even $N$ is $ℰ\mathcal{(}N):=R_{a,4}\left(N\right)-M_{a,4}\left(N\right)$.

Remark 4. The enclosure $G\in [1.4132088648,1.4132089899]$ is verified by computing the partial product to $P={10}^5$ and bounding the tail via ${\sum }_{p>P}^{}{(p-1)}^{-2}<8.86\times {10}^{-8}$. The constant $K\le 38.02$ is the final value in Paper 1 , obtained with $G_{lo}=1.4132088648$. An earlier draft of Paper 1 used the more conservative bound $K\le 38.82$ (corresponding to $G_{hi}=1.4132089899$); the present paper adopts the tighter value $K\le 38.02$ throughout.

The Structural Obstruction

In this section we prove that eliminating GRH from Theorem 2 is equivalent to proving GRH. This is not a deficiency of the two papers but a fundamental feature of the Hardy–Littlewood circle method applied pointwise.

Proposition 5 (Structural obstruction). Suppose there exists a constant ${\beta }_0\in (\frac{1}{2},1)$ and a Dirichlet character $\chi$ modulo $4$ such that $L(s,\chi )$ has a non-trivial zero ${\rho }_0={\beta }_0+i{\gamma }_0$. Then there exist a constant $c>0$ and an infinite sequence of even integers $N_k\to \infty$ such that:

$\mathcal{|}ℰ\mathcal{(}{N}_k)|\ge c\hspace{0.17em}{N}_k^{{\beta }_0}$ for all $k$;

$N_k$ does not belong to the exceptional set of Theorem 1 for $A<2{\beta }_0-1$;

$N_k$ violates the pointwise bound of Theorem 2.

Proof. By the explicit formula applied to $\psi (x,\chi )$ and to the convolution defining $R_{a,4}\left(N\right)$, the error decomposes as

The contribution of the conjugate pair $\{{\rho }_0,{\bar{\rho }}_0\}$ to $ℰ\mathcal{(}N)$ is

By Weyl’s equidistribution theorem, the sequence $\{{\gamma }_{0}logN/\left(2\pi \right){\}}_{N\hspace{0.17em}\mathrm{\text{even}}}$ is equidistributed modulo 1, so there exists an infinite subsequence $N_k$ of even integers with $\left|cos({\gamma }_{0}log{N}_k+arg{\rho }_0)\right|\ge \frac{1}{2}$. For those $N_k$,

Claim (i) is immediate.

Claim (ii). The exceptional set of Theorem 1 at level $A$ consists of $N$ for which $\mathcal{|}ℰ\mathcal{(}N)|>C(A,4)\hspace{0.17em}N{\left(logN\right)}^{-3}$. For large $k$ and ${\beta }_0\in (\frac{1}{2},1)$, we have $N_k^{{\beta }_0}\ll N_k^{1-\delta }$ for $\delta =1-{\beta }_0>0$, while $C(A,4)\hspace{0.17em}{N}_k{\left(log{N}_k\right)}^{-3}$ grows like $N_k$ (faster than $N_k^{{\beta }_0}$ since ${\beta }_0<1$). Hence for any fixed $A$ and sufficiently large $k$, $\mathcal{|}ℰ\mathcal{(}{N}_k)|=c{N}_k^{{\beta }_0}<C(A,4)N_k{\left(log{N}_k\right)}^{-3}$, so $N_k$ is not in the exceptional set of Paper 1. (This is the key asymmetry: the Paper 1 exceptional set captures $N$ with very large error, not the moderate error $N^{{\beta }_0}$ with ${\beta }_0<1$.)

Claim (iii). Since ${\beta }_0>\frac{1}{2}$, for any $\epsilon <{\beta }_0-\frac{1}{2}$ and large $k$, $\mathcal{|}ℰ\mathcal{(}{N}_k)|>N_k^{1/2+\epsilon }$, which violates the pointwise bound of Theorem 2. ◻

Corollary 6 (Equivalence with GRH). The Following are Equivalent:

For every even $N\ge {10}^{19.9}$, $R_{a,4}\left(N\right)>0$.

GRH holds for all Dirichlet $L$-functions modulo $4$.

Proof. $\left(b\right)\Rightarrow \left(a\right)$ is Theorem 2.

$\left(a\right)\Rightarrow \left(b\right)$: Contrapositively, if GRH fails, then by Proposition 5 there are infinitely many even $N_k\ge {10}^{19.9}$ with $\mathcal{|}ℰ\mathcal{(}{N}_k)|\ge c{N}_k^{{\beta }_0}$. Since $M_{a,4}\left(N\right)\asymp N/logN\ll N^{{\beta }_0}$ is eventually dominated by the error, $R_{a,4}\left(N_k\right)=M_{a,4}\left(N_k\mathcal{)}\mathcal{+}ℰ\mathcal{(}{N}_k\right)$ can be zero (or negative, which is impossible by definition, hence zero) for a subsequence of $N_k$. ◻

Remark 7. Corollary 6 makes precise the sense in which the conditional hypothesis of Paper 2 is necessary, not merely sufficient. No strengthening of Paper 1’s circle-method argument can remove GRH from Paper 2.

Main Unconditional Result

Key Lemmas From Papers 1 and 2 [1, 2]

Lemma 8 (From Paper 1 [1]). For any $A>0$ there exist effective constants $C(A,4)$ and $E_1(A,4)$ such that $\#\{N\le X\hspace{0.17em}\mathcal{:}\hspace{0.17em}\mathcal{|}ℰ\mathcal{(}N)|>C(A,4)\hspace{0.17em}N{\left(logN\right)}^{-3}\}\hspace{0.17em}\le \hspace{0.17em}{E}_1(A,4)\hspace{0.17em}\frac{X}{{\left(logX\right)}^A}.$

Lemma 9 (From Paper 2 [2], Sections 6–7). For every even $N\ge {10}^{19.9}$, $M_{a,4}\left(N\right)\hspace{0.17em}\ge \hspace{0.17em}\frac{C_2\hspace{0.17em}N}{2logN}.$ Moreover, if $R_{a,4}\left(N\right)=0$, then $ℰ\mathcal{(}N)=-M_{a,4}\left(N\right)$, so $\mathcal{|}ℰ\mathcal{(}N)|=M_{a,4}\left(N\right)\hspace{0.17em}\ge \hspace{0.17em}\frac{C_2\hspace{0.17em}N}{2logN}.$

Proof. Since $S\left(N\right)\ge 1$ for even $N$ and $\phi \left(4\right)=2$, we have $M_{a,4}\left(N\right)=C_{2}S\left(N\right)N/2\ge C_{2}N/2$. The stated bound $C_{2}N/\left(2logN\right)$ is conservative; the full main term satisfies $M_{a,4}\left(N\right)\gg N/logN$ with an effective constant. ◻

The Main Theorem

Theorem 10 (Main unconditional bound). Let $q=4$, $a\in \{1,3\}$. For any $A>0$ and all $X\ge X_0\left(A\right):=max({10}^{20},e^{47})$,

For $A=1$ the proportion of potential counterexamples among all even integers in $[{10}^{19.9},X]$ is at most $E_1(1,4)/logX$.

Proof. Let $𝔈\mathfrak{(}X):=\{N\in [{10}^{19.9},X]:N\hspace{0.17em}\mathrm{\text{even}},\hspace{0.17em}{R}_{a,4}\left(N\right)=0\}$.

Step 1 (Error is large). If $N\mathfrak{\in }𝔈\mathfrak{(}X)$, then $R_{a,4}\left(N\right)=0$, so $ℰ\mathcal{(}N)=-M_{a,4}\left(N\right)$ and by Lemma 9,

Step 2 (Threshold comparison). Choose $C(A,4)$ as in Lemma 8. We claim that for $N\ge N_1\left(A\right)\le {10}^{20}$,

This is equivalent to

which holds for all $N\ge exp\left(\sqrt{2C(A,4)/C_2}\right)$.

For $A=1$: $C(1,4)\le 19.01$, so the threshold is

which is well below ${10}^{20}$.

Numerical verification at $N={10}^{20}$ (where $logN=46.0517\dots$):

The inequality $LHS>RHS$ holds with a factor of $\approx 36.7$.

Step 3 (Containment in exceptional set). From Steps 1 and 2, every $N\mathfrak{\in }𝔈\mathfrak{(}X)$ with $N\ge N_1\left(A\right)$ satisfies $\mathcal{|}ℰ\mathcal{(}N)|>C(A,4)\hspace{0.17em}N{\left(logN\right)}^{-3}$, hence lies in the exceptional set of Lemma 8.

Step 4 (Counting). Applying Lemma 8 to the range $[{10}^{19.9},X]$:

The lower endpoint ${10}^{19.9}$ contributes only a finite constant (bounded by $X_0\left(A\right)$), absorbed into $E_1$. ◻

Remark 11 (Explicit constant for $A=1$). For $A=1$, an explicit conservative estimate gives $E_1(1,4)\le E(1,4)$ were

The value $E(1,4)\approx 2.3\times {10}^3$ quoted elsewhere is a more conservative upper bound including additional factors from the second-moment proof. Any finite value suffices for the asymptotic statement.

Quantitative Interpretation and Comparison with Pintz

What the Theorem Says and Does not Say

Observation 12 (Precise scope of Theorem 10).

Numerical Illustration

For $X={10}^{30}$ and $A=1$:

The fraction of even integers in the range that could fail Goldbach is at most $E(1,4)/log\left({10}^{30}\right)\approx E(1,4)/69$. For $A=10$: the fraction drops to ${\left(logX\right)}^{-10}\approx {10}^{-18}$.

Table 1 gives numerical values of the two competing bounds for selected values of $X$.

Genuine Paths Toward an Unconditional Pointwise Result

We record the three analytical strategies that could eventually yield $R_{a,4}\left(N\right)>0$ for all $N\ge N_1$ unconditionally. None is within reach of present techniques, but each is strictly weaker than GRH.

Via the Density Hypothesis $𝐃𝐇\mathbf{\left(}\mathrm{𝟐}\mathbf{\right)}$

Under $DH\left(2\right)$ (the standard density hypothesis $N(\sigma ,T,\chi )\ll T^{2(1-\sigma )+\epsilon }$), Paper 2 gives $\theta \left(2\right)=1/2$ and every even $N\ge N_1(2,4)$ satisfies $R_{a,4}\left(N\right)>0$. The Density Hypothesis is strictly weaker than GRH ($\forall$ zeros on $\sigma =1/2$ vs. a density bound). Known unconditional values: Huxley gives $A=12/5$, $\theta =6/11\approx 0.545$; Bourgain gives $A\approx 2.78$; neither reaches $A=2$.

Via the Chen-type Unconditional Theorem

Theorem 9.1 of Paper 2 [2]  (proved unconditionally via the Selberg sieve and Bombieri–Vinogradov) gives: for every even $N\ge N_{Chen}\left(4\right)\le e^{50}$, $N=p+P_2$ with $p\equiv a\left(mod4\right)$ and $\omega \left(P_2\right)\le 2$. This is a genuine pointwise unconditional result, but $P_2$ is not necessarily prime. Selberg’s parity barrier prevents upgrading $\omega \left(P_2\right)\le 2$ to $\omega \left(P_2\right)=1$ by sieve methods alone.

Via Computational Verification

The empirical verification scripts accompanying Papers 1 and 2 [1, 2]  implement a deterministic Miller–Rabin primality test (valid for $n<3.317\times {10}^{24}$) with a symmetric search from $N/2$. Results:

• Exhaustive sieve verification: $R_{a,4}\left(N\right)>0$ for all even $N\le {10}^7$, zero failures.

• Random sampling: 20,000 samples per zone in $[4\times {10}^{18},{10}^{23}]$, zero failures across all six zones.

• Individual verification: all ten test cases in $[4\times {10}^{18},{10}^{24}]$ confirmed positive in $<72$ ms each.

Closing the gap $[4\times {10}^{18},{10}^{19.9}]$ exhaustively would require $7.5\times {10}^{19}$ primality tests; at ${10}^{9}N/s$ with ${10}^4$ cores this takes $\approx 2,378$ years.

DISCUSSION AND SUMMARY

What is and is not A Proof of Goldbach

Theorem 10 is an unconditional result about the density of potential counterexamples, not about their existence. A proof of the binary Goldbach conjecture (even restricted to $q=4$) would require showing $R_{a,4}\left(N\right)>0$ for every even $N\ge N_1$ — a pointwise statement. Corollary 6 shows this is equivalent to GRH within the framework of Papers 1 and 2. Our result closes the question of what is achievable by combining these two papers without additional hypotheses.

Is This Result Binary or Ternary Goldbach?

This paper concerns exclusively the binary Goldbach problem $N=p_1+p_2$ with the restriction $p_1\equiv a\left(mod4\right)$. The ternary analogue ($N=p_1+p_2+p_3$) is not treated here; it has been settled unconditionally for all odd $N>5$ (Helfgott 2013), and the restricted ternary version follows by a standard transfer from the binary almost-all result using the Chen-type theorem of Paper 2.

The Novelty of the Combination

The contribution of this paper is not in any single new analytic estimate but in the logical analysis of what the combination of Papers 1 and 2 [1, 2] can and cannot yield. Specifically:

The structural obstruction (Corollary 6) is new: it shows the conditional hypothesis of Paper 2 is logically necessary for any pointwise result combining the two papers.

The density bound (Theorem 10) is new as a combined consequence: neither Paper 1 alone (which gives no information above the almost-all threshold) nor Paper 2 alone (which is conditional) yields an unconditional density bound in $[{10}^{19.9},X]$.

The quantitative comparison with Pintz (Proposition 13) is new for the restricted $q=4$ setting and clarifies the relationship between the two types of bounds.

CONCLUSION

We have established three results concerning the restricted binary Goldbach sum $R_{a,4}\left(N\right)$ in the range $N\ge {10}^{19.9}$:

Structural obstruction (unconditional). Extracting a pointwise result $R_{a,4}\left(N\right)>0$ for all $N\ge {10}^{19.9}$ from the combination of Papers 1 and 2 is equivalent to GRH for all Dirichlet $L$-functions modulo 4. This is a genuine impossibility result within the circle-method framework.

Density theorem (unconditional). For any $A>0$ and all $X\ge {10}^{20}$:

In particular, the fraction of potential counterexamples in $[{10}^{19.9},X]$ tends to zero faster than any power of $logX$. For $A=1$, $K\le 38.02$, the constant satisfies $E_1(1,4)\lesssim 586$ (conservative: $\lesssim 2.3\times {10}^3$).

Comparison with Pintz. The density conclusion in 2 is stronger than the Pintz bound $O\left(X^{0.72}\right)$ in relative density over $[{10}^{19.9},X]$: the ratio $X^{0.28}/{\left(logX\right)}^A\to +\infty$ for any $A>0$.

These results do not prove the Goldbach conjecture. They identify precisely what can and cannot be proved by combining the almost-all theorem of Paper 1 [1] with the GRH-conditional threshold of Paper 2 [2], and establish the best unconditional density statement extractable from that combination