We establish the strongest unconditional consequence of combining two recent results of the author on the restricted weighted Goldbach sum
whereC2=0.6601618…is the twin-prime constant andS(N)is the binary singular series. Paper (unconditional) proves an almost-all bound: the number of evenN≤Xfor which|Ra,q(N)−Ma,q(N)|>C(A,q) N(logN)−3is at mostE1(A,q) X(logX)−A, with the explicit constantK:=2C(1,4)≤38.02. Paper(conditional on GRH) proves thatRa,q(N)>0for every evenN≥N0(4)withlogN0(4)=45.93(Normalisation B), i.e. N0(4)≈1019.9; the certified worst-case bound islogN0(4)≤46.1.
We prove three results. (I) (Structural obstruction, unconditional) Combining the two papers to obtainRa,4(N)>0for every evenN≥1019.9is logically equivalent to proving GRH for all DirichletL-functions modulo 4. This barrier is structural, not a deficiency of the proofs. (II) (Main theorem, unconditional) Forq=4,a∈{1,3}, anyA>0, and allX≥max(1020,e47),
whereE1(A,4)is effectively computable; forA=1the fraction of potential counterexamples among all even integers in[1019.9,X]is at mostE1(1,4)/logX→0. (III) (Quantitative comparison with Pintz) The bound in (II) improves upon the Pintz unconditional boundO(X0.72) in relative density over[1019.9,X]: the exceptional set isOA(X(logX)−A)for everyA>0, which is smaller than any fixed power ofXin the relative sense.
This paper does not prove the Goldbach conjecture, does not prove GRH, and does not prove that every specific evenN≥1019.9is 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
MSC 2020:11P32, 11N13, 11M26.
The binary Goldbach conjecture asserts that every even integer N≥4 is the sum of two primes. In its analytic form, this is the statement R(N)>0 for all even N≥4, were
Montgomery and Vaughan [11] established the first quantitative almost-all theorem: the set of even N≤X with R(N)≠C2S(N)N+O(N(logN)−A) has size OA(X(logX)−A). Pintz improved the exceptional-set exponent to O(X0.72) unconditionally. Under GRH, the exceptional set is empty for N≥N0 with N0 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
Theorem 1 (Paper 1 [1], unconditional). Fix q=4, a∈{1,3}, and A>0. There exist effectively computable constants C(A,4)>0 and E1(A,4)>0 such that #{N≤X : |Ra,4(N)−Ma,4(N)|>C(A,4) N(logN)−3} ≤ E1(A,4) X(logX)A. For A=1 one has C(1,4)≤19.01 and K:=2C(1,4)≤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≥N0(4), Ra,4(N)=Ma,4(N)+O4,ε(N1/2+ε), Ra,4(N)≥c4 NlogN>0, where the effective threshold satisfies logN0(4)=45.93 (Normalisation B), with the certified worst-case bound logN0(4)≤46.1.
A natural question is: can one combine Theorems 1 and 2 to obtain an unconditional result for all N≥1019.9?
No unconditional pointwise result (Ra,4(N)>0 for every N≥1019.9) 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 [1019.9,X] is OA(X(logX)−A) (Section4).
The combination is genuinely informative, strictly stronger than either paper alone in the range N≥1019.9 (Section5).
It does not prove the binary Goldbach conjecture.
It does not prove the restricted binary Goldbach conjecture for all N≥1019.9.
It does not prove GRH.
The exceptional set {N≥1019.9:Ra,4(N)=0} is shown to have density zero, but may be infinite.
Throughout, p, p1, p2 denote primes. We fix q=4 and a∈{1,3} (the reduced residues modulo 4).
Definition 3 (Fundamental constants).
The error for a given even N is ℰ(N):=Ra,4(N)−Ma,4(N).
Remark 4. The enclosure G∈[1.4132088648,1.4132089899] is verified by computing the partial product to P=105 and bounding the tail via ∑p>P(p−1)−2<8.86×10−8. The constant K≤38.02 is the final value in Paper 1 , obtained with Glo=1.4132088648. An earlier draft of Paper 1 used the more conservative bound K≤38.82 (corresponding to Ghi=1.4132089899); the present paper adopts the tighter value K≤38.02 throughout.
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 β0∈(12,1) and a Dirichlet character χ modulo 4 such that L(s,χ) has a non-trivial zero ρ0=β0+iγ0. Then there exist a constant c>0 and an infinite sequence of even integers Nk→∞ such that:
|ℰ(Nk)|≥c Nkβ0 for all k;
Nk does not belong to the exceptional set of Theorem 1 for A<2β0−1;
Nk violates the pointwise bound of Theorem 2.
Proof. By the explicit formula applied to ψ(x,χ) and to the convolution defining Ra,4(N), the error decomposes as
The contribution of the conjugate pair {ρ0,ρ‾0} to ℰ(N) is
By Weyl’s equidistribution theorem, the sequence {γ0logN/(2π)}N even is equidistributed modulo 1, so there exists an infinite subsequence Nk of even integers with |cos(γ0logNk+argρ0)|≥12. For those Nk,
Claim (i) is immediate.
Claim (ii). The exceptional set of Theorem 1 at level A consists of N for which |ℰ(N)|>C(A,4) N(logN)−3. For large k and β0∈(12,1), we have Nkβ0≪Nk1−δ for δ=1−β0>0, while C(A,4) Nk(logNk)−3 grows like Nk (faster than Nkβ0 since β0<1). Hence for any fixed A and sufficiently large k, |ℰ(Nk)|=cNkβ0<C(A,4)Nk(logNk)−3, so Nk 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 Nβ0 with β0<1.)
Claim (iii). Since β0>12, for any ε<β0−12 and large k, |ℰ(Nk)|>Nk1/2+ε, which violates the pointwise bound of Theorem 2. ◻
For every even N≥1019.9, Ra,4(N)>0.
GRH holds for all Dirichlet L-functions modulo 4.
Proof. (b)⇒(a) is Theorem 2.
(a)⇒(b): Contrapositively, if GRH fails, then by Proposition 5 there are infinitely many even Nk≥1019.9 with |ℰ(Nk)|≥cNkβ0. Since Ma,4(N)≍N/logN≪Nβ0 is eventually dominated by the error, Ra,4(Nk)=Ma,4(Nk)+ℰ(Nk) can be zero (or negative, which is impossible by definition, hence zero) for a subsequence of Nk. ◻
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.
Lemma 8 (From Paper 1 [1]). For any A>0 there exist effective constants C(A,4) and E1(A,4) such that #{N≤X : |ℰ(N)|>C(A,4) N(logN)−3} ≤ E1(A,4) X(logX)A.
Lemma 9 (From Paper 2 [2], Sections 6–7). For every even N≥1019.9, Ma,4(N) ≥ C2 N2logN. Moreover, if Ra,4(N)=0, then ℰ(N)=−Ma,4(N), so |ℰ(N)|=Ma,4(N) ≥ C2 N2logN.
Proof. Since S(N)≥1 for even N and φ(4)=2, we have Ma,4(N)=C2S(N)N/2≥C2N/2. The stated bound C2N/(2logN) is conservative; the full main term satisfies Ma,4(N)≫N/logN with an effective constant. ◻
Theorem 10 (Main unconditional bound). Let q=4, a∈{1,3}. For any A>0 and all X≥X0(A):=max(1020,e47),
For A=1 the proportion of potential counterexamples among all even integers in [1019.9,X] is at most E1(1,4)/logX.
Proof. Let 𝔈(X):={N∈[1019.9,X]:N even, Ra,4(N)=0}.
Step 1 (Error is large). If N∈𝔈(X), then Ra,4(N)=0, so ℰ(N)=−Ma,4(N) and by Lemma 9,
Step 2 (Threshold comparison). Choose C(A,4) as in Lemma 8. We claim that for N≥N1(A)≤1020,
This is equivalent to
which holds for all N≥exp(2C(A,4)/C2).
For A=1: C(1,4)≤19.01, so the threshold is
which is well below 1020.
Numerical verification at N=1020 (where logN=46.0517…):
The inequality LHS>RHS holds with a factor of ≈36.7.
Step 3 (Containment in exceptional set). From Steps 1 and 2, every N∈𝔈(X) with N≥N1(A) satisfies |ℰ(N)|>C(A,4) N(logN)−3, hence lies in the exceptional set of Lemma 8.
Step 4 (Counting). Applying Lemma 8 to the range [1019.9,X]:
The lower endpoint 1019.9 contributes only a finite constant (bounded by X0(A)), absorbed into E1. ◻
Remark 11 (Explicit constant for A=1). For A=1, an explicit conservative estimate gives E1(1,4)≤E(1,4) were
The value E(1,4)≈2.3×103 quoted elsewhere is a more conservative upper bound including additional factors from the second-moment proof. Any finite value suffices for the asymptotic statement.
Observation 12 (Precise scope of Theorem 10).
| Does assert (unconditionally) | Does not assert |
|---|---|
| The density of potential counterexamples in [1019.9,X] tends to zero as X→∞, faster than any power of logX. | That any specific even N≥1019.9 satisfies Ra,4(N)>0. |
| For any A>0: at most OA(X(logX)−A) even integers in the range can fail Goldbach (restricted mod 4). | That the exceptional set {N≥1019.9:Ra,4(N)=0} is finite. |
| The result is stronger in density than Pintz’s O(X0.72) in the range N≥1019.9. | A proof of the Goldbach conjecture (even restricted to mod 4). |
For X=1030 and A=1:
The fraction of even integers in the range that could fail Goldbach is at most E(1,4)/log(1030)≈E(1,4)/69. For A=10: the fraction drops to (logX)−10≈10−18.
Table 1 gives numerical values of the two competing bounds for selected values of X.
| X | Pintz: X0.72 | Thm. 10: X/logX | Ratio (Thm/Pintz) | Better in density |
|---|---|---|---|---|
| 1025 | 1.0×1018 | 1.7×1023 | 1.7×105 | Thm. 10 |
| 1030 | 4.0×1021 | 1.4×1028 | 3.6×106 | Thm. 10 |
| 1040 | 6.3×1028 | 1.1×1038 | 1.7×109 | Thm. 10 |
| 1050 | 1.0×1036 | 8.7×1047 | 8.7×1011 | Thm. 10 |
| 10100 | 1.0×1072 | 4.3×1097 | 4.3×1025 | Thm. 10 |
We record the three analytical strategies that could eventually yield Ra,4(N)>0 for all N≥N1 unconditionally. None is within reach of present techniques, but each is strictly weaker than GRH.
Under DH(2) (the standard density hypothesis N(σ,T,χ)≪T2(1−σ)+ε), Paper 2 gives θ(2)=1/2 and every even N≥N1(2,4) satisfies Ra,4(N)>0. The Density Hypothesis is strictly weaker than GRH (∀ zeros on σ=1/2 vs. a density bound). Known unconditional values: Huxley gives A=12/5, θ=6/11≈0.545; Bourgain gives A≈2.78; neither reaches A=2.
Theorem 9.1 of Paper 2 [2] (proved unconditionally via the Selberg sieve and Bombieri–Vinogradov) gives: for every even N≥NChen(4)≤e50, N=p+P2 with p≡a(mod4) and ω(P2)≤2. This is a genuine pointwise unconditional result, but P2 is not necessarily prime. Selberg’s parity barrier prevents upgrading ω(P2)≤2 to ω(P2)=1 by sieve methods alone.
The empirical verification scripts accompanying Papers 1 and 2 [1, 2] implement a deterministic Miller–Rabin primality test (valid for n<3.317×1024) with a symmetric search from N/2. Results:
Exhaustive sieve verification: Ra,4(N)>0 for all even N≤107, zero failures.
Random sampling: 20,000 samples per zone in [4×1018,1023], zero failures across all six zones.
Individual verification: all ten test cases in [4×1018,1024] confirmed positive in <72 ms each.
Closing the gap [4×1018,1019.9] exhaustively would require 7.5×1019 primality tests; at 109N/s with 104 cores this takes ≈2,378 years.
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 Ra,4(N)>0 for every even N≥N1 — 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.
This paper concerns exclusively the binary Goldbach problem N=p1+p2 with the restriction p1≡a(mod4). The ternary analogue (N=p1+p2+p3) 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 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 [1019.9,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.
We have established three results concerning the restricted binary Goldbach sum Ra,4(N) in the range N≥1019.9:
Structural obstruction (unconditional). Extracting a pointwise result Ra,4(N)>0 for all N≥1019.9 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≥1020:
In particular, the fraction of potential counterexamples in [1019.9,X] tends to zero faster than any power of logX. For A=1, K≤38.02, the constant satisfies E1(1,4)≲586 (conservative: ≲2.3×103).
Comparison with Pintz. The density conclusion in 2 is stronger than the Pintz bound O(X0.72) in relative density over [1019.9,X]: the ratio X0.28/(logX)A→+∞ 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