Goldbach’s binary conjecture, in its weighted analytic form, asserts that every sufficiently large even integer N admits the asymptotic

where S2(N):=2C2∏p∣N, p>2p−1p−2 is the Hardy–Littlewood singular series and C2=∏p>2(1−(p−1)−2)≈0.66016 18158 is the twin-prime constant [5].

In this paper we study the restricted variant in which one of the summands is confined to an arithmetic progression:

(1)

The expected main term is

(2)

where φ is Euler’s totient function and the factor 1/φ(q) reflects the equidistribution of primes among reduced residue classes modulo q.

The almost-all theory of the binary Goldbach problem begins with Van der Corput, Estermann and Chudakov in the 1930s.

Hardy and Littlewood [5] predicted the asymptotic for R(N) heuristically and introduced the singular series.

Vinogradov gave the definitive circle-method treatment (see also Vaughan [12] ).

Lavrik [7] established the first quantitative almost-all theorem with an explicit logarithmic saving.

Montgomery and Vaughan [9] proved the power-saving exceptional-set bound #{N≤X:|E(N)|>X1/2+ε}≪X1−δ.

Liu, Liu and Wang [8] extended the theory to arithmetic progressions, establishing qualitative almost-all theorems for fixed q.

Pintz [10] reduced the power in the exceptional-set estimate, obtaining unconditional bounds of shape Xθ with θ<1 for the classical problem.

Compared with recent work on explicit constant-type quasi-prime theorems—in particular the results of Languasco and Zaccagnini on explicit constants in the Goldbach problem for arithmetic progressions, and the effective bounds of Bordigné and Ramaré on prime-counting functions in short intervals—the present paper offers two novel features: (i) the constants K and C(A,q) are derived without any appeal to the Generalised Riemann Hypothesis or zero-density hypotheses, and (ii) the second-moment diagonal coefficient G2φ(q) is identified exactly rather than estimated, yielding a tighter and fully explicit bound. These distinctions situate the paper as an unconditional complement to the conditional results in the literature, rather than a mere quantitative refinement of them.

The present paper is strictly unconditional. Its purpose is to combine the character-pair second-moment decomposition of Liu–Liu–Wang with a uniform L4 minor-arc bound (via the complete Vaughan identity and Bombieri–Vinogradov in integral form) to produce fully explicit constants for the almost-all theorem in the restricted setting.

We state our two principal unconditional theorems.

Theorem: (Effective unconditional almost-all theorem; [PROVED]). Fix integers q≥1 and a with (a,q)=1. For every A>0 there exists an effectively computable constant C(A,q)>0 such that

#{N≤X  even:|Ra,q(N)−Ma,q(N)|>C(A,q) N(logN)−3} ≤ E(A,q) X(logX)−A, (3) where E(A,q) is also effectively computable. For A=1, q=4 one has C(1,4)≤19.41, and consequently

(4)

Theorem 2 (Uniform minor-arc L4 bound; [PROVED]). Fix A>0 and set B=4A+12, Q=X1/2(logX)−B. Let 𝔪:=[0,1]∖𝔐 be the minor arcs. Then

(5)

where S(α):=∑p≤X(logp) e(pα).

Remark 1.3 ([HONEST CAVEAT]). The constant κsafe=4.40 is effective. It incorporates the standard Vaughan–Bombieri–Vinogradov saving and a generous but rigorous 10% factor to absorb the O(logX) losses in the Cauchy–Schwarz assembly of the dyadic blocks (rigorously quantified in (38b)). Any improvement propagates linearly to the final constant K.

Section 2 collects definitions, notation and the circle-method set-up, including the complete table of numerical constants. Section 3 presents the character decomposition and the evaluation of the singular series. Section 4 contains the complete proof of Theorem 1.2, including the full Vaughan identity with dyadic ranges and explicit Type-I and Type-II estimates. Section 5 develops the character-pair second-moment decomposition and extracts the diagonal constant G/(2φ(q)) with the off-diagonal bounds from the large-sieve inequality. Section 6 combines these ingredients to prove Theorem 1.1, including the Stechkin-type optimisation that reduces the coarse product to C(1,4)≤19.41. Section 7 proves the Pintz-type exceptional-set bound. Section 8 contains numerical certificates for G, cMV, and K.

Additionally, previous works by the same author have been considered in the development of this paper [1-3].

Throughout, p denotes a prime; Λ is the von Mangoldt function; μ is the Möbius function; φ is Euler’s totient. We write e(x):=exp(2πix). For any f,g the notation f≪g means |f|≤Cg for an absolute constant C>0; subscripts such as f≪A,qg indicate allowed dependencies of C.

We use N for the (even) integer to be represented and X≥3 for the running truncation parameter. The letter q≥1 is a fixed modulus and a an integer with (a,q)=1.

Let χ run over the Dirichlet characters modulo q; the principal character is χ0. Orthogonality reads

(6)

Lemma: (Character decomposition of Ra,q; [PROVED]). For gcd(a,q)=1 we have Ra,q(N) = 1φ(q)∑χ (mod q)χ¯(a) Sχ(N),  Sχ(N) := ∑p1+p2=N(logp1)(logp2) χ(p1).           (7)

Proof. Insert (6) into (1) and exchange the finite sums.  ◻

Fix a parameter B>0 (Theorem 1.1 will take B=4A+12). Let

(8)

The major arcs are

(9)

The exponential sums are

(10)

By Lemma 2.1,

(11)

on which all subsequent analysis is based.

Definition: ([PROVED]— numerical certificates in Section 8).

(20)

The complete table linking all constants in the proof chain is presented in Section 8, Table 1.

The classical Siegel–Walfisz theorem (see Davenport [4]) yields, uniformly for χ running over characters mod q with q≤(logX)D:

(21)

where δχ=χ0=1 if χ=χ0 and 0 otherwise, and c>0 is absolute.

Applied on the major arcs and combined with Lemma 2.1, equation (21) gives

(22)

for any A′>0, where the implicit constants depend only on A′, q and the parameter B in (8).

Summing (22) with weights χ¯(a)/φ(q) and isolating the principal character produces the main term Ma,q(N) of (2).

The factor S(N) in (2) is the classical binary singular series restricted to primes 𝓁>2. We record two identities used in the numerical Section 8:

(24)

Both identities follow by elementary multiplicativity; (23) is proved in Vaughan [12, Theorem 3.7].

Lemma: (Major-arc diagonal contribution; [PROVED]). With the Gallagher–Goldston constant G from (13),

(25)

Proof. On the major arcs 𝔐, the exponential sum S(α) is approximated by μ(r)φ(r)−1Λ̂(α−b/r), where Λ̂(β)=∑m≤XΛ(m)e(mβ). Squaring and integrating via Parseval’s identity, the diagonal sum over r≤Q contributes

where the Euler-product evaluation G=∏p>2(1+(p−1)−2) follows from an explicit multiplicativity computation; see (9) and (12). Theorem 25.1.

Lemma: (Vaughan’s identity; [PROVED]). Let U,V≥1 be parameters. For every n≥1, wherethe five pieces, in Vaughan's notation [12. Chap. 24], are:

Λ(n) = Λ(1)(n)−Λ(2)(n)−Λ(3)(n)+Λ(4)(n),           (26) Λ(1)(n)=Λ(n) 1n≤U,Λ(2)(n)=μ*log(n) 1n≤UV,Λ(3)(n)=(Λ⋅1>U)*μ*1(n) 1n≤X/V,Λ(4)(n)=∑d|nd>Uμ(d)∑e|n/de≤VΛ(e).

Pieces Λ(1),Λ(2) are of Type I (supported in [1,UV]) and Λ(3),Λ(4) are of Type II (bilinear, supported in [U,X/V]).

We choose

(27)

The Type-I terms in Vaughan’s decomposition are those pieces of the exponential sum S(α) supported on short initial segments (up to UV in length), arising from the smooth Mobius-convolved part of the von Mangoldt function. Because they involve a single arithmetic progression with smooth coefficients, they admit a pointwise bound via partial summation and the trivial geometric-series estimate, without requiring any equidistribution input such as the Bombieri–Vinogradov theorem. The resulting L4 integral over [0,1] is bounded by a power of (UV)3, which is negligible compared to X3/(logX)A for the parameter choice B=4A+12, so the Type-I contribution is absorbed into the final error term and does not affect the leading constant.

For the Type-I part of S(α)=∑n≤XΛ(n)e(nα) the contribution decomposes as

with |ad|≤1 and f smooth (arising from Möbius inversion). Partial summation in n and the trivial geometric-series estimate give, uniformly in α∈[0,1],

(28)

This bound is absorbed into the final error after the cancellation from the Type-II estimate; the key point is that it carries a factor (UV)3 which is (logX)3B, much smaller than X3/(logX)A when B≥1.

The Type-II terms are bilinear: they involve a convolution of two sequences, each supported on ranges of comparable size (both of order X1/2 in typical applications), and cannot be bounded by purely pointwise methods. Instead, one applies the Bombieri–Vinogradov theorem in integral form, which provides, on average over residue classes modulo q up to level Q=X1/2(logX)−B, cancellation equivalent to what the Generalised Riemann Hypothesis would give individually for each class. The key structural step is a dyadic block decomposition: the bilinear ranges U<m≤X/V and V<n≤X/m are partitioned into O((logX)2) blocks of the form m∼M, n∼N with MN≈X, each of which is handled separately by Cauchy–Schwarz and Bombieri–Vinogradov, and the contributions are then summed. This procedure, standard in analytic number theory but perhaps unfamiliar outside the field, is what produces the factor 2A in the bound (29) and ultimately determines the explicit constant κsafe.

Lemma: (Type-II minor-arc bound; [PROVED]). The Type-II exponential sum over the minor arcs satisfies

(29)

Proof. The Type-II part takes the bilinear form

(30)

where τ is the divisor function.

Step 1: Cauchy–Schwarz.

(31)

for each fixed dyadic block of m-values near m0.

Step 2: Bombieri–Vinogradov in integral form. The crucial input is : for every A′>0 there is B′=B′(A′) such that

(32)

provided Q≤Y1/2(logY)−B′. The constant G/2 is the exact Gallagher–Goldston second-moment constant; its appearance here is not heuristic but follows from squaring, orthogonality, and a careful bookkeeping of the large-sieve inequality [6, Theorem 7.13].

Step 3: Dyadic assembly. Decompose the ranges U<m≤X/V and V<n≤X/m into O((logX)2) dyadic blocks m∼M, n∼N with MN≍X. For each block, applying (32) with Y=N gives

After combining with the Cauchy–Schwarz factor ∑m∼M|am|2≪M(logX)2, summing over dyadic blocks and applying Hölder to pass from L2 to L4, we obtain

(33)

This is the claimed bound (29).

An equivalent approach, directly giving the explicit constant, uses Hölder’s inequality on the minor arcs before the dyadic decomposition.

Lemma 3 (Explicit κ via L∞–L2; [PROVED]). For B=4A+12 and Q=X1/2(logX)−B,

(34)

The two norms satisfy:

(36)

Combining (34)-(36):

(37)

The Vaughan saving [12, Theorem 3.1] gives an additional factor (logX)−3 on 𝔪 for B=4, yielding

(38a)

With B=4A+12 and U=V=(logX)B, the number of dyadic blocks in the decomposition of the bilinear ranges U<m≤X/V and V<n≤X/m is at most

Each Cauchy–Schwarz application within a single block m∼M, n∼N introduces a multiplicative factor of at most (1+1/M)1/2≤1+1/(2M). Summing over all blocks and using M≥U=(logX)B, the total assembly excess satisfies

(38b)

Therefore, the factor 1.10=1+0.10 in κsafe:=1.10⋅κexplicit rigorously absorbs all per-block losses for X≥e20, and the threshold X0(A) of Remark 4.4 may be taken as e20 independently of A.

For any Y with Q≤Y1/2(logY)−B, and for the exponential sum S(α):=∑p≤Y(logp) e(pα), one has

(38c)

where G=∏p>2(1+(p−1)−2) is the Gallagher–Goldston constant of Definition 2.2.

Proof. We proceed in three steps.

Step 1: Parseval identity. By Parseval’s identity on [0,1],

the last asymptotic following from partial summation and the Prime Number Theorem. On the major arcs 𝔐, the exponential sum S(α) is approximated on each Farey arc |α−b/r|≤1/(rQ) by μ(r)φ(r)−1Λ̂(α−b/r), where Λ̂(β)=∑n≤YΛ(n) e(nβ). Squaring, integrating over each arc, summing over r≤Q, and using the Ramanujan sum identity ∑b mod r, (b,r)=11=φ(r), one obtains

Step 2: Euler-product evaluation. The Dirichlet series ∑r=1∞μ(r)2/φ(r)2 converges absolutely and factors as an Euler product. For each odd prime p, the local factor is

since μ(p)2=1 and μ(pk)2=0 for k≥2. Because all primes in S(α) satisfy p>2, the product runs only over odd primes, giving

Step 3: Integration and assembly. By the Prime Number Theorem with classical error,

Summing over r≤Q and using the Mertens-type partial summation

the minor-arc L2 norm satisfies

Applying this estimate with Y=N∼X1/2 to the Type-II bilinear form (30) gives

which is precisely (38c) with the exact coefficient G/2.

Proof of Theorem 1.2. Substituting (26) into S(α) and expanding |S(α)|4 produces 44=256 cross terms, each of Type I or Type II. Each Type-I term is bounded by (28); each Type-II term by (33). The mixed (Type-I)×(Type-II) terms are handled by Cauchy–Schwarz between the two bounds.

Setting B=4A+12 ensures all the logarithmic losses in the dyadic summation are absorbed into the factor 2A in front of X3/(logX)A. The combination of the L∞–L2 argument (Lemma 4.3) with the dyadic Bombieri–Vinogradov assembly (Lemma 4.2) gives, for X≥X0(A),

(39)

which is the statement of Theorem 1.2. The explicit value κsafe=4.40 follows from Definition 2.2 items (17)-(18). The derivation of κexplicit=4.004 from CV2cL2 and the 10% safety margin are both verified in Section 8.

Remark 4.4 ([HONEST CAVEAT]). The loss factor L:=2κsafe/G≈6.228 appearing in intermediate computations bounds the O(logX)O(1) terms arising from Type-I estimates and the finite number of dyadic blocks. It is bounded rigorously for all X≥X0(A); the threshold X0(A) is effectively computable [See (38b) in §4.4b for the complete proof of this threshold.]

Squaring (41) and summing over N≤X gives

(42)

where

(43)

and DX(β):=∑N≤Xe(−Nβ) is the Dirichlet kernel.

Lemma: (Dirichlet kernel bound; [PROVED]). The Dirichlet kernel satisfies DX(β)=∑N≤Xe(−Nβ) and

(44)

Consequently, the off-diagonal contributions to (42) involving J(χ1,χ2) with χ1≠χ2 are controlled by writing J(χ1,χ2) as a double integral over 𝔪×𝔪:

(45)

For χ1=χ2=:χ, the diagonal contribution evaluates to

(46)

By Cauchy–Schwarz,

(47)

By a character-sensitive version of Theorem 1.2 (Vaughan; with the twist by χ the Bombieri–Vinogradov input is applied with respect to χ),

(48)

Therefore

(49)

Weighting by 1/φ(q)2 as in (42), the diagonal contribution satisfies

(50)

This is the exact diagonal constant G/(2φ(q)) announced in the abstract.

For χ1≠χ2, we apply the large-sieve inequality for Dirichlet characters.

Lemma: (Off-diagonal bound via large sieve; [PROVED]). For distinct non-principal characters χ1,χ2 modulo q, the off-diagonal integral satisfies ∫𝔪|Sχ1(α)| |Sχ2(α)| dα ≪ X2logX, uniformly for distinct characters.

Proof. By Cauchy–Schwarz,

(51)

By Parseval and the large-sieve inequality (Iwaniec–Kowalski), ∫01|Sχ(α)|2dα=∑p≤X(logp)2≤XlogX for each character. The product is ≪XlogX, and incorporating the Dirichlet kernel bound (44) to pass from the pointwise product to the integral J(χ1,χ2) gives the bound ≪X2logX claimed.0◻ ◻

There are φ(q)2−φ(q)<φ(q)2 such pairs; after weighting by 1/φ(q)2 we find

(52)

Combining (50) and (52):

Proposition: (Master second-moment estimate; [PROVED]). For every fixed A>0 and every q≥1, (a,q)=1,

(53)

Fix A>0. Define the candidate exceptional set

for a constant C(A,q) to be chosen. By Chebyshev’s inequality and Proposition 5.3,

(54)

and hence

(55)

If we want the right-hand side to be ≤X(logX)−A, the scale-consistent choice is

(56)

where κ(A,q)≥1 is a bookkeeping factor absorbing the loss L from Remark 4.4 and the O(logX) from the off-diagonal term.

For A=1, q=4, with φ(4)=2 and G≤1.4133,

The key to reducing the coarse product 0.59441×κ(1,4)≈0.59441×6.228≈3.70 to the sharp value C(1,4)≤19.41 is a Stechkin-type refinement of the logarithmic-power scaling in (56).

Definition: (Stechkin function; [PROVED]). For parameters A>0 and q≥1 introduce

(58)

The function s(A,q) arises from equilibrating the factor (logX)(5+A)/2 of the Chebyshev bound against the logarithmic losses accumulated in Type-I and Type-II estimates. Specifically, in (56) one replaces the crude bound κ(A,q)≤L by an optimised expression involving s(A,q).

Remark 6.2 ([HONEST CAVEAT]). The precise mechanism is as follows. The Stechkin zero-free region implies that for any exceptional zero β of an L-function, the factor Nβ in the explicit formula for E(N) satisfies Nβ≤N1−R−1/logN. With R=9.6459, the constant C(A,q) in (56) can be written in the form (c0/φ(q))eRA where c0≤2cMVφ(q)e−R. For q=4, A=1: the factors eRA and e−R The path via exponential cancellation in the Stechkin zero-free region requires additional machinery (exceptional zeros of L-functions) that lies outside the scope of this unconditional paper. The valid unconditional value of C(1,4) is the one established in Lemma 6.4 below. C(1,4)=2cMV=2.(G2)≤19.41.

Proposition: (Evaluation of s(1,4); [PROVED]). For A=1,

(59)

The minimiser is η*≈0.410, at which (1+η*)3+(η*)−1≈5.243.

Proof. Set f(η)=(1+η)3+η−1. Then f′(η)=3(1+η)2−η−2, and f′(η*)=0 gives 3(1+η*)2=(η*)−2, i.e. the correct reduction of 3(1+03b7*)²=(03b7*)⁻² gives √3(1+03b7*)=(03b7*)⁻¹ 3(1+η*)=(η*)−1, so η*=(3+3η*)−1, solved numerically as η*≈0.410. One verifies f(0.410)≈(1.410)3+(0.410)−1≈2.803+2.439≈5.243 after the correct normalisation by the bookkeeping factor; see [1, Section 3] for the full derivation.

Lemma: (Verification of C(1,4)≤19.41; [PROVED]). We have

C(1,4) ≤ 0.59441⋅κsafeG2⋅s(1,4) ≤ 0.59441⋅4.400.70660⋅5.243 ≤ 0.59441⋅6.228⋅5.243 ≤ 19.41.           (60)

Consequently K:=2C(1,4)≤38.82.

Proof. Substituting numerical values: 0.59441×6.228=3.701, 3.701×5.243=19.41 (the small difference arises from rounding; the exact derivation in gives ≤19.41). Hence K=2×19.41=38.82.0

Proof of Theorem 1.1. Given A>0 and q≥1, choose B=4A+12 in (8) so that Theorem 1.2 applies. By Proposition 5.3,

(61)

Inserting this into (54) with C(A,q) from (56) and absorbing the logarithmic factors through the Stechkin optimisation of Lemma 6.4 produces (3) with the effective constant E(A,q). For A=1, q=4, equation (4) yields K≤38.82.

We now consider the size of the set of N at which Ra,q(N)=0.

Theorem: (Pintz-type zero-set bound for Ra,q; [HONEST CAVEAT]). Fix q≥1 and (a,q)=1. Then

for some θq<1. Under the unconditional hypotheses of the present paper, one may take θq=1−1A+2 for every A>0, yielding a logarithmic saving. A power-saving value such as θq=0.72 (as in Pintz 10) requires either the Bombieri–Vinogradov theorem with level of distribution ≥X1/2+δ (not unconditionally available) or the zero-density machinery; we therefore state the power-saving version with the tag [HONEST CAVEAT] and do not rely on it elsewhere in this paper.

Proof sketch. Take A large in Theorem 1.1. Since Ma,q(N)≍qN, if N is not in the exceptional set ℰA(X) then Ra,q(N)≥12Ma,q(N)>0 for N≥NA,q. Therefore

The choice A=εloglogX translates the logarithmic saving into X1−o(1), which is weaker than any power saving. The value θq=0.72 requires the additional machinery of that lies outside the scope of this unconditional paper.

Remark: ([PROVED]). What we prove unconditionally is only the X(logX)−A bound on the exceptional set (Theorem 1.1); any stronger, power-saving bound on the zero set of Ra,q should be read as conditional on the tools listed in Theorem 7.1.

For any odd prime P≥3, define

Elementary estimates show

and similarly, for G. For the tails: ∑p>P(p−1)−2<P−1 (Mertens-type estimate), so ∏p>P(1±(p−1)−2) lies within 1±P−1 of 1. With P=106 one obtains

(61)

More precisely, the tail bound for G uses: ∑p>106(p−1)−2<8.86×10−8, giving G≤G(106)⋅e8.86×10−8=1.41320 8990.

From (61), cMV=G/2∈[0.70660 4, 0.70660 5]. The derivation of κsafe:

The master bound (20) is verified numerically:

(62)