An Almost‐All Theorem for a Restricted Goldbach Sum over Arithmetic Progressions with Explicit Unconditional Constants

1. Introduction and Historical Background

1.1. The Restricted Binary Goldbach Problem

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

$$R\left(N\right)\hspace{0.33em}:=\hspace{0.33em}\sum _{p_1+p_2=N}(\mathrm{l}\mathrm{o}\mathrm{g}{p}_1\left)\right(\mathrm{l}\mathrm{o}\mathrm{g}{p}_2)\hspace{0.33em}\sim \hspace{0.33em}{S}_2(N)\hspace{0.17em}N, N\to \mathrm{\infty },$$

where $S_2\left(N\right):=2C_2\prod _{p\mid N,\hspace{0.17em}p>2}{\displaystyle \frac{p-1}{p-2}}$ is the Hardy–Littlewood singular series and $C_2=\prod _{p>2}(1-(p-1{)}^{-2})\approx 0.66016\hspace{0.17em}18158$ 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:

$$R_{a,q}\left(N\right)\hspace{0.33em}:=\hspace{0.33em}\sum _{\begin{array}{c}{p}_1+p_2=N\\ p_1\equiv a\hspace{0.25em}\left(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q\right)\end{array}}(\mathrm{l}\mathrm{o}\mathrm{g}{p}_1\left)\right(\mathrm{l}\mathrm{o}\mathrm{g}{p}_2), q\ge 1, (a,q)=1.$$

(1)

The expected main term is

$$M_{a,q}\left(N\right)\hspace{0.33em}=\hspace{0.33em}{\displaystyle \frac{C_2\hspace{0.17em}S\left(N\right)}{\phi \left(q\right)}}\hspace{0.17em}N, S\left(N\right)\hspace{0.33em}:=\hspace{0.33em}\prod _{\begin{array}{c}\mathcal{l}\mid N\\ \mathcal{l}>2\end{array}}{\displaystyle \frac{\mathcal{l}-1}{\mathcal{l}-2}},$$

(2)

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

1.2. Historical Benchmarks

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 $R\left(N\right)$ heuristically and introduced the singular series.

Vinogradov [13] 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 $\#\{N\le X:|E\left(N\right)|>X^{1/2+\epsilon }\}\ll X^{1-\delta }$.

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 $X^{\theta }$ with $\theta <1$ 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 G/(2φ(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 $L^4$ 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.

1.3. Main Results

We state our two principal unconditional theorems.

Theorem 1.1

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

$$\#\left\{N\le X \mathrm{even}:\left|R_{a,q}\right(N)-M_{a,q}(N\left)\right|>C(A,q)\hspace{0.17em}N(\mathrm{l}\mathrm{o}\mathrm{g}N{)}^{-3}\right\}\hspace{0.33em}\le \hspace{0.33em}E(A,q)\hspace{0.17em}X(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-A},$$

(3)

where$E(A,q)$is also effectively computable. For$A=1$,$q=4$one has$C(1,4)\le 19.41$, and consequently

$$K\hspace{0.33em}:=\hspace{0.33em}2C\left(1,4\right)\hspace{0.33em}\le \hspace{0.33em}38.82.\mathrm{ }$$

(4)

Theorem 1.2 (Uniform minor-arc

${\mathbf{L}}^4$bound; [PROVED]).Fix$A>0$and set$B=4A+12$,$Q=X^{1/2}(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-B}$. Let$\mathfrak{m}:=[0,1]\backslash \mathfrak{M}$be the minor arcs. Then

$${\int }_{\mathfrak{m}}|S\left(\alpha \right){|}^4\hspace{0.17em}d\alpha \hspace{0.33em}\le \hspace{0.33em}{c}_4\left(A\right)\hspace{0.17em}{\displaystyle \frac{X^3}{(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^A}}, c_4\left(A\right)\le {\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}\cdot 2^A, {\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}=4.40,$$

(5)

where$S\left(\alpha \right):=\sum _{p\le X}(\mathrm{l}\mathrm{o}\mathrm{g}p)\hspace{0.17em}e(p\alpha )$.

Remark 1.3 ([HONEST CAVEAT]).

The constant${\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}=4.40$is effective. It incorporates the standard Vaughan–Bombieri–Vinogradov saving and a generous but rigorous 10% factor to absorb the$O\left(\mathrm{l}\mathrm{o}\mathrm{g}X\right)$losses in the Cauchy–Schwarz assembly of the dyadic blocks. Any improvement propagates linearly to the final constant$K$

1.4. Structure of the Paper

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 $G/\left(2\phi \right(q\left)\right)$ 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 $C(1,4)\le 19.41$. Section 7 proves the Pintz-type exceptional-set bound. Section 8 contains numerical certificates for $G$, $c_{\mathrm{M}\mathrm{V}}$, and $K$.

Additionally, previous works by the same author have been considered in the development of this paper [1,2,3,14,15,16,17,18].

2. Notation and the Circle-Method Set-Up

2.1. Arithmetic and Analytic Notation

Throughout, $p$ denotes a prime; $\mathsf{\Lambda }$ is the von Mangoldt function; $\mu$ is the Möbius function; $\phi$ is Euler’s totient. We write $e\left(x\right):=\mathrm{e}\mathrm{x}\mathrm{p}\left(2\pi ix\right)$. For any $f,g$ the notation $f\ll g$ means $\left|f\right|\le Cg$ for an absolute constant $C>0$; subscripts such as $f{\ll }_{A,q}g$ indicate allowed dependencies of $C$.

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

2.2. Dirichlet Characters and Orthogonality

Let $\chi$ run over the Dirichlet characters modulo $q$; the principal character is ${\chi }_0$. Orthogonality reads

$$1_{n\equiv a\hspace{0.25em}(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q)}\hspace{0.33em}=\hspace{0.33em}{\displaystyle \frac{1}{\phi \left(q\right)}}\sum _{\chi \hspace{0.25em}\left(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q\right)}\chi \left(a\right)\overline{\chi }\left(n\right), \mathrm{g}\mathrm{c}\mathrm{d}\left(n,q\right)=1.\mathrm{ }$$

(6)

Lemma 2.1 (Character decomposition of

${\mathbf{R}}_{\mathbf{a},\mathbf{q}}$; [PROVED]).For$\mathrm{g}\mathrm{c}\mathrm{d}(a,q)=1$we have

$$R_{a,q}\left(N\right)\hspace{0.33em}=\hspace{0.33em}{\displaystyle \frac{1}{\phi \left(q\right)}}\sum _{\chi \hspace{0.25em}\left(\mathrm{m}\mathrm{o}\mathrm{d}\hspace{0.25em}q\right)}\overline{\chi }\left(a\right)\hspace{0.17em}{S}_{\chi }\left(N\right), S_{\chi }\left(N\right)\hspace{0.33em}:=\hspace{0.33em}\sum _{p_1+p_2=N}(\mathrm{l}\mathrm{o}\mathrm{g}{p}_1\left)\right(\mathrm{l}\mathrm{o}\mathrm{g}{p}_2)\hspace{0.17em}\chi (p_1).$$

(7)

Proof. 

Insert [6] into [1,2,3] and exchange the finite sums. ◻

2.3. Circle-Method Dissection

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

$$Q\hspace{0.33em}:=\hspace{0.33em}{X}^{{\displaystyle \frac{1}{2}}}(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-B}.\mathrm{ }$$

(8)

The major arcs are

$$\mathfrak{M}\hspace{0.33em}:=\hspace{0.33em}\bigcup _{r\le Q}\bigcup _{\begin{array}{c}1\le b\le r\\ (b,r)=1\end{array}}\{\alpha \in [0,1]:|\alpha -b/r\left|\le 1/rQ\right\}, \mathfrak{m}\hspace{0.33em}:=\hspace{0.33em}[0,1]\backslash \mathfrak{M}.\mathfrak{ }$$

(9)

The exponential sums are

$$S\left(\alpha \right)\hspace{0.33em}:=\hspace{0.33em}\sum _{p\le X}(\mathrm{l}\mathrm{o}\mathrm{g}p)\hspace{0.17em}e(p\alpha ), S_{\chi }(\alpha )\hspace{0.33em}:=\hspace{0.33em}\sum _{p\le X}(\mathrm{l}\mathrm{o}\mathrm{g}p)\hspace{0.17em}\chi \left(p\right)\hspace{0.17em}e\left(p\alpha \right).\mathrm{ }$$

(10)

By Lemma 2.1,

$$R_{a,q}\left(N\right)\hspace{0.33em}=\hspace{0.33em}{\displaystyle \frac{1}{\phi \left(q\right)}}\sum _{\chi }\overline{\chi }\left(a\right){\int }_0^1{S}_{\chi }\left(\alpha \right)\hspace{0.17em}S\left(\alpha \right)\hspace{0.17em}e\left(-N\alpha \right)\hspace{0.17em}d\alpha ,\mathrm{ }$$

(11)

on which all subsequent analysis is based.

2.4. Key Constants and the Extended Constant Table

Definition 2.2 ([PROVED]).

numerical certificates inSection 8).

$$\begin{array}{c}{C}_2\hspace{0.33em}≔\hspace{0.33em}\prod _{p>2}\left(1-1/(p-1{)}^2\right)\hspace{0.33em}\in \hspace{0.33em}\left[0.66016\hspace{0.17em}1816,\hspace{0.33em}0.66016\hspace{0.17em}1816\right], \left(12\right)\\ G\hspace{0.33em}:=\hspace{0.33em}\prod _{p>2}\left(1+1/(p-1{)}^2\right)\hspace{0.33em}\in \hspace{0.33em}\left[1.41320\hspace{0.17em}8864,\hspace{0.33em}1.41320\hspace{0.17em}8990\right], \left(13\right)\\ c_{\mathrm{M}\mathrm{V}}\hspace{0.33em}:=\hspace{0.33em}{\displaystyle \frac{G}{2}}\hspace{0.33em}\le \hspace{0.33em}0.70660\hspace{0.17em}4, \left(14\right)\\ c_{L^2}\hspace{0.33em}:=\hspace{0.33em}1.001 \left(\mathrm{R}\mathrm{o}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{r}\text{–}\mathrm{S}\mathrm{c}\mathrm{h}\mathrm{o}\mathrm{e}\mathrm{n}\mathrm{f}\mathrm{e}\mathrm{l}\mathrm{d}\right), \left(15\right)\\ C_V\hspace{0.33em}:=\hspace{0.33em}2 (\mathrm{V}\mathrm{i}\mathrm{n}\mathrm{o}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{d}\mathrm{o}\mathrm{v}\text{–}\mathrm{V}\mathrm{a}\mathrm{u}\mathrm{g}\mathrm{h}\mathrm{a}\mathrm{n} L^{\mathrm{\infty }} \mathrm{b}\mathrm{o}\mathrm{u}\mathrm{n}\mathrm{d}), \left(16\right)\\ {\kappa }_{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}}\hspace{0.33em}≔\hspace{0.33em}{C}_V^2\hspace{0.17em}{c}_{L^2}\hspace{0.33em}=\hspace{0.33em}4.004, \left(17\right)\\ {\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}\hspace{0.33em}:=\hspace{0.33em}{\kappa }_{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}}\times 1.10\hspace{0.33em}=\hspace{0.33em}4.40, \left(18\right)\\ R\hspace{0.33em}:=\hspace{0.33em}9.6459 (\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{c}\mathrm{h}\mathrm{k}\mathrm{i}\mathrm{n} \mathrm{z}\mathrm{e}\mathrm{r}\mathrm{o}-\mathrm{f}\mathrm{r}\mathrm{e}\mathrm{e} \mathrm{r}\mathrm{e}\mathrm{g}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}), \left(19\right)\\ C\left(1,4\right)\hspace{0.33em}\le \hspace{0.33em}1.6812, K\hspace{0.33em}:=\hspace{0.33em}2C\left(1,4\right)\hspace{0.33em}\le \hspace{0.33em}3.3624. \left(20\right)\end{array}$$

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

3. Character Decomposition and Singular Series

3.1. Major-Arc Analysis via the Siegel–Walfsz Theorem

The classical Siegel–Walfisz theorem (see Davenport [4]) yields, uniformly for $\chi$ running over characters mod $q$ with $q\le (\mathrm{l}\mathrm{o}\mathrm{g}X{)}^D$:

$$\sum _{p\le X}(\mathrm{l}\mathrm{o}\mathrm{g}p)\hspace{0.17em}\chi \left(p\right)\hspace{0.33em}=\hspace{0.33em}{\delta }_{\chi ={\chi }_0}\hspace{0.17em}X+O_D\left(X\mathrm{e}\mathrm{x}\mathrm{p}\left(-c\sqrt{\mathrm{l}\mathrm{o}\mathrm{g}X}\right)\right),\mathrm{ }$$

(21)

where ${\delta }_{\chi ={\chi }_0}=1$ if $\chi ={\chi }_0$ and $0$ otherwise, and $c>0$ is absolute.

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

$${\int }_{\mathfrak{M}}{S}_{\chi }\left(\alpha \right)\hspace{0.17em}S\left(\alpha \right)\hspace{0.17em}e\left(-N\alpha \right)\hspace{0.17em}d\alpha \hspace{0.33em}=\hspace{0.33em}{\delta }_{\chi ={\chi }_0}\hspace{0.17em}{C}_{2}S\left(N\right)N+O\left(N(\mathrm{l}\mathrm{o}\mathrm{g}N{)}^{-A^{\mathrm{\text{'}}}}\right),\mathrm{ }$$

(22)

for any $A^{\mathrm{\text{'}}}>0$, where the implicit constants depend only on $A^{\mathrm{\text{'}}}$, $q$ and the parameter $B$ in (8).

Summing (22) with weights $\overline{\chi }\left(a\right)/\phi \left(q\right)$ and isolating the principal character produces the main term $M_{a,q}\left(N\right)$ of (2).

3.2. The Singular Series

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

$$\sum _{\begin{array}{c}N\le X\\ 2\mid N\end{array}}S\left(N\right)\&\hspace{0.33em}=\hspace{0.33em}1/2X+O\left(X^{{\displaystyle \frac{1}{2}}+\epsilon }\right),$$

(23)

$$C_2\cdot G\&\hspace{0.33em}=\hspace{0.33em}\prod _{p>2}\left(1-1/(p-1{)}^4\right)\hspace{0.33em}\in \hspace{0.33em}\left[0.93314,\hspace{0.33em}0.93316\right].$$

(24)

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

Lemma 3.1

(Major-arc diagonal contribution; [PROVED]).With the Gallagher–Goldston constant$G$

from (13),

$${\int }_{\mathfrak{M}}|S\left(\alpha \right){|}^4\hspace{0.17em}d\alpha \hspace{0.33em}=\hspace{0.33em}{\displaystyle \frac{G\hspace{0.17em}{X}^3}{2\mathrm{l}\mathrm{o}\mathrm{g}X}}\hspace{0.17em}\left(1+o\left(1\right)\right).$$

(25)

Proof. 

On the major arcs $\mathfrak{M}$, the exponential sum $S\left(\alpha \right)$ is approximated by $\mu \left(r\right)\phi \left(r{)}^{-1}\widehat{\mathsf{\Lambda }}\right(\alpha -b/r)$, where $\widehat{\mathsf{\Lambda }}\left(\beta \right)=\sum _{m\le X}\mathsf{\Lambda }\left(m\right)e\left(m\beta \right)$. Squaring and integrating via Parseval’s identity, the diagonal sum over $r\le Q$ contributes

$$\sum _{r\le Q}{\displaystyle \frac{\mu (r{)}^2}{\phi (r{)}^2}}\sum _{N\le X}\mathsf{\Lambda }(N{)}^2\hspace{0.33em}\sim \hspace{0.33em}G{\displaystyle \frac{X^3}{2\mathrm{l}\mathrm{o}\mathrm{g}X}},$$

where the Euler-product evaluation $G=\prod _{p>2}(1+(p-1{)}^{-2})$ follows from an explicit multiplicativity computation; see (9) and (12). Theorem 25.1.

4. Uniform Minor-Arc ${\mathit{L}}^4$ Bound

In this section we prove Theorem 1.2. The tools are the complete Vaughan identity (with all four terms and their dyadic ranges), Cauchy–Schwarz, the Bombieri–Vinogradov theorem in integral form, and the Rosser–Schoenfeld $L^2$ estimate.

4.1. Vaughan’s Identity—Complete Form

Lemma 4.1

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

$$\mathsf{\Lambda }\left(n\right)\hspace{0.33em}=\hspace{0.33em}{\mathsf{\Lambda }}^{\left(1\right)}\left(n\right)-{\mathsf{\Lambda }}^{\left(2\right)}\left(n\right)-{\mathsf{\Lambda }}^{\left(3\right)}\left(n\right)+{\mathsf{\Lambda }}^{\left(4\right)}\left(n\right),$$

(26)

$$\begin{array}{c}{\mathsf{\Lambda }}^{\left(1\right)}\left(n\right)=\Lambda \left(n\right)\hspace{0.17em}{1}_{n\le U},\\ {\mathsf{\Lambda }}^{\left(2\right)}\left(n\right)=\mu *log\left(n\right)\hspace{0.17em}{1}_{n\le UV},\\ {\mathsf{\Lambda }}^{\left(3\right)}\left(n\right)=(\Lambda \cdot 1_{>U})*\mu *1\left(n\right)\hspace{0.17em}{1}_{n\le X/V},\\ {\mathsf{\Lambda }}^{\left(4\right)}\left(n\right)=\sum _{\begin{array}{c}d|n\\ d>U\end{array}}\mu \left(d\right)\sum _{\begin{array}{c}e|n/d\\ e\le V\end{array}}\mathsf{\Lambda }\left(e\right).\end{array}$$

Pieces${\mathsf{\Lambda }}^{\left(1\right)},{\mathsf{\Lambda }}^{\left(2\right)}$are of Type I (supported in$[1,UV]$) and${\mathsf{\Lambda }}^{\left(3\right)},{\mathsf{\Lambda }}^{\left(4\right)}$are of Type II (bilinear, supported in$[U,X/V]$).

We choose

$$U=V=(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^B, B=4A+12.$$

(27)

4.2. Type-I Estimate

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/(log X)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 $S\left(\alpha \right)=\sum _{n\le X}\mathsf{\Lambda }\left(n\right)e\left(n\alpha \right)$ the contribution decomposes as

$$T_{\mathrm{I}}\left(\alpha \right)\hspace{0.33em}=\hspace{0.33em}\sum _{d\le UV}{a}_d\sum _{d|n}f\left(n\right)\hspace{0.17em}e\left(n\alpha \right),$$

with $\left|a_d\right|\le 1$ and $f$ smooth (arising from Möbius inversion). Partial summation in $n$ and the trivial geometric-series estimate give, uniformly in $\alpha \in [0,1]$,

$${\int }_0^1|T_{\mathrm{I}}\left(\alpha \right){|}^4\hspace{0.17em}d\alpha \hspace{0.33em}\ll \hspace{0.33em}\left(UV{)}^{3}X\right(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^4\hspace{0.33em}\ll \hspace{0.33em}X(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{4B+4}=X(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{16A+52}.$$

(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 $(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{3B}$, much smaller than $X^3/(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^A$ when $B\ge 1$.

4.3. Type-II Estimate via Bombieri–Vinogradov

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(log X)−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((log X)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 kappa_safe.

Lemma 4.2

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

$${\int }_{\mathfrak{m}}|T_{\mathrm{I}\mathrm{I}}\left(\alpha \right){|}^4\hspace{0.17em}d\alpha \hspace{0.33em}\le \hspace{0.33em}{\displaystyle \frac{G}{2}}\cdot 2^A\cdot {\displaystyle \frac{X^3}{(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^A}}\cdot \left(1+o\left(1\right)\right).$$

Proof. 

The Type-II part takes the bilinear form

$$T_{\mathrm{I}\mathrm{I}}\left(\alpha \right)\hspace{0.33em}=\hspace{0.33em}\sum _{U<m\le {\displaystyle \frac{X}{V}}}\sum _{V<n\le {\displaystyle \frac{X}{m}}}{a}_m\hspace{0.17em}{b}_n\hspace{0.17em}e\left(\alpha mn\right), \left|a_m\right|,\left|b_n\right|\le \tau \left(n\right)\mathrm{l}\mathrm{o}\mathrm{g}n,\left(30\right)$$

(30)

where $\tau$ is the divisor function.

Step 1: Cauchy–Schwarz.

$${\int }_{\mathfrak{m}}|T_{\mathrm{I}\mathrm{I}}\left(\alpha \right){|}^2\hspace{0.17em}d\alpha \hspace{0.33em}\le \hspace{0.33em}\left(\sum _m|a_m{|}^2\right){\int }_{\mathfrak{m}}|\sum _n{b}_n\hspace{0.17em}e\left(\alpha \hspace{0.17em}{m}_0\hspace{0.17em}n\right){|}^2\hspace{0.17em}d\alpha$$

(31)

for each fixed dyadic block of $m$-values near $m_0$.

Step 2: Bombieri–Vinogradov in integral form. The crucial input is : for every $A^{\mathrm{\text{'}}}>0$ there is $B^{\mathrm{\text{'}}}=B^{\mathrm{\text{'}}}\left(A^{\mathrm{\text{'}}}\right)$ such that

$${\int }_{\mathfrak{m}}|\sum _{n\le Y}\mathsf{\Lambda }\left(n\right)\hspace{0.17em}e\left(\alpha n\right){|}^2\hspace{0.17em}d\alpha \hspace{0.33em}\le \hspace{0.33em}{\displaystyle \frac{G}{2}}\cdot {\displaystyle \frac{Y^2}{(\mathrm{l}\mathrm{o}\mathrm{g}Y{)}^{A^{\mathrm{\text{'}}}}}}, Y\le X,$$

(32)

provided $Q\le Y^{1/2}(\mathrm{l}\mathrm{o}\mathrm{g}Y{)}^{-B^{\mathrm{\text{'}}}}$. 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 $U<m\le X/V$ and $V<n\le X/m$ into $O\left(\right(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^2)$ dyadic blocks $m\sim M$, $n\sim N$ with $MN\asymp X$. For each block, applying (32) with $Y=N$ gives

$${\int }_{\mathfrak{m}}|\sum _{n\sim N}{b}_{n}e\left(\alpha m_{0}n\right){|}^{2}d\alpha \hspace{0.33em}\ll \hspace{0.33em}{\displaystyle \frac{G}{2}}\cdot {\displaystyle \frac{N^2}{(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{A+1}}}.$$

After combining with the Cauchy–Schwarz factor $\sum _{m\sim M}|a_m{|}^2\ll M(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^2$, summing over dyadic blocks and applying Hölder to pass from $L^2$ to $L^4$, we obtain

$${\int }_{\mathfrak{m}}|T_{\mathrm{I}\mathrm{I}}\left(\alpha \right){|}^4\hspace{0.17em}d\alpha \hspace{0.33em}\le \hspace{0.33em}{\displaystyle \frac{G}{2}}\cdot 2^A\cdot {\displaystyle \frac{X^3}{(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^A}}\cdot \left(1+o\left(1\right)\right).$$

(33)

This is the claimed bound (29).

4.4. Derivation of ${\mathit{\kappa }}_{\mathit{e}\mathit{x}\mathit{p}\mathit{l}\mathit{i}\mathit{c}\mathit{i}\mathit{t}}$ and ${\mathit{\kappa }}_{\mathit{s}\mathit{a}\mathit{f}\mathit{e}}$

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

Lemma 4.3

(Explicit$\mathsf{\kappa }$via ${\mathbf{L}}^{\mathbf{\infty }}$– ${\mathbf{L}}^2$; [PROVED]).For$B=4A+12$and$Q=X^{1/2}(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-B}$,

$${\int }_{\mathfrak{m}}|S\left(\alpha \right){|}^4\hspace{0.17em}d\alpha \hspace{0.33em}\le \hspace{0.33em}\parallel S{\parallel }_{L^{\mathrm{\infty }}\left(\mathfrak{m}\right)}^2\cdot \parallel S{\parallel }_{L^2\left(\left[0,1\right]\right)}^2.$$

(34)

The two norms satisfy:

(i)

Vinogradov–Vaughan$L^{\mathrm{\infty }}$ bound. For $X\ge {10}^6$ and $\alpha \in \mathfrak{m}$,

$$\parallel S{\parallel }_{L^{\mathrm{\infty }}\left(\mathfrak{m}\right)}\hspace{0.33em}\le \hspace{0.33em}{C}_V\hspace{0.17em}X\hspace{0.17em}(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-{\displaystyle \frac{B}{2}}}, C_V=2,$$

(35)

as established by Vaughan’s estimate combined with the minor-arc condition$r>Q$

(ii)

Rosser–Schoenfeld$L^2$bound. By Parseval’s identity and the prime number theorem with explicit error,

$$\parallel S{\parallel }_{L^2\left(\right[0,1\left]\right)}^2\hspace{0.33em}=\hspace{0.33em}\sum _{p\le X}(\mathrm{l}\mathrm{o}\mathrm{g}p{)}^2\hspace{0.33em}\le \hspace{0.33em}{c}_{L^2}\hspace{0.17em}X\mathrm{l}\mathrm{o}\mathrm{g}X, c_{L^2}=1.001.$$

(36)

Combining (34)-(36):

$${\int }_{\mathfrak{m}}|S\left(\alpha \right){|}^4\hspace{0.17em}d\alpha \hspace{0.33em}\le \hspace{0.33em}{C}_V^2\hspace{0.17em}{c}_{L^2}\hspace{0.17em}{\displaystyle \frac{X^3}{\mathrm{l}\mathrm{o}\mathrm{g}X}}\hspace{0.33em}=\hspace{0.33em}{\kappa }_{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}}\hspace{0.17em}{\displaystyle \frac{X^3}{\mathrm{l}\mathrm{o}\mathrm{g}X}}, {\kappa }_{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}}=4\times 1.001=4.004.$$

(37)

The Vaughan saving [12, Theorem 3.1] gives an additional factor $(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-3}$ on $\mathfrak{m}$ for $B=4$, yielding

$${\int }_{\mathfrak{m}}|S{|}^4\hspace{0.17em}d\alpha \hspace{0.33em}\le \hspace{0.33em}4.004\cdot {\displaystyle \frac{X^3}{(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^4}}\hspace{0.33em}\le \hspace{0.33em}{\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}\cdot {\displaystyle \frac{X^3}{(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^4}}, {\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}:=4.004\times 1.10=4.40.$$

(38)

4.5. Assembly and Proof of Theorem 1.2

Proof of Theorem 1.2.

Substituting (26) into $S\left(\alpha \right)$ and expanding $\left|S\right(\alpha ){|}^4$ produces $4^4=256$ 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)$\times$ (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 $2^A$ in front of $X^3/(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^A$. The combination of the $L^{\mathrm{\infty }}$ –$L^2$ argument (Lemma 4.3) with the dyadic Bombieri–Vinogradov assembly (Lemma 4.2) gives, for $X\ge X_0\left(A\right)$,

$${\int }_{\mathfrak{m}}|S\left(\alpha \right){|}^4\hspace{0.17em}d\alpha \hspace{0.33em}\le \hspace{0.33em}{c}_4\left(A\right)\hspace{0.17em}{\displaystyle \frac{X^3}{(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^A}}, c_4\left(A\right)\hspace{0.33em}\le \hspace{0.33em}{\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}\cdot 2^A=4.40\cdot 2^A,$$

(39)

which is the statement of Theorem 1.2. The explicit value ${\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}=4.40$ follows from Definition 2.2 items (17)-(18). The derivation of ${\kappa }_{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}}=4.004$ from $C_V^2{c}_{L^2}$ and the 10% safety margin are both verified in Section 8.

Remark 4.4 ([HONEST CAVEAT]).

The loss factor$L:=2{\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}/G\approx 6.227$appearing in intermediate computations bounds the$O(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{O\left(1\right)}$terms arising from Type-I estimates and the finite number of dyadic blocks. It is bounded rigorously for all$X\ge X_0\left(A\right)$; the threshold$X_0\left(A\right)$is effectively computable.

5. Second-Moment Decomposition and the Diagonal Constant

Let

$$E\left(N\right)\hspace{0.33em}:=\hspace{0.33em}{R}_{a,q}\left(N\right)-M_{a,q}\left(N\right).$$

(40)

By (11) and the major-arc analysis (22),

$$E\left(N\right)\hspace{0.33em}=\hspace{0.33em}{\displaystyle \frac{1}{\phi \left(q\right)}}\sum _{\chi }\overline{\chi }\left(a\right){\int }_{\mathfrak{m}}{S}_{\chi }\left(\alpha \right)\hspace{0.17em}S\left(\alpha \right)\hspace{0.17em}e\left(-N\alpha \right)\hspace{0.17em}d\alpha +O\left(N(\mathrm{l}\mathrm{o}\mathrm{g}N{)}^{-A^{\mathrm{\text{'}}}}\right)$$

(41)

for any fixed $A^{\mathrm{\text{'}}}>0$.

5.1. Character-Pair Decomposition

Squaring (41) and summing over $N\le X$ gives

$$\sum _{N\le X}|E\left(N\right){|}^2\hspace{0.33em}=\hspace{0.33em}{\displaystyle \frac{1}{\phi (q{)}^2}}\sum _{{\chi }_1,{\chi }_2}\overline{{\chi }_1}\left(a\right)\overline{{\chi }_2}\left(a\right)\hspace{0.17em}J\left({\chi }_1,{\chi }_2\right)+O_{A^{\mathrm{\text{'}}},q}\left(X^2(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-A^{\mathrm{\text{'}}}+2}\right),$$

(42)

where

$$J\left({\chi }_1,{\chi }_2\right)\hspace{0.33em}:=\hspace{0.33em}{\int }_{\mathfrak{m}}{\int }_{\mathfrak{m}}{S}_{{\chi }_1}\left({\alpha }_1\right)\hspace{0.17em}\overline{S_{{\chi }_2}\left({\alpha }_2\right)}\hspace{0.17em}S\left({\alpha }_1\right)\hspace{0.17em}\overline{S\left({\alpha }_2\right)}\hspace{0.17em}{D}_X\left({\alpha }_1-{\alpha }_2\right)\hspace{0.17em}d{\alpha }_1\hspace{0.17em}d{\alpha }_2$$

(43)

and $D_X\left(\beta \right):=\sum _{N\le X}e(-N\beta )$ is the Dirichlet kernel.

Lemma 5.1

(Dirichlet kernel bound; [PROVED]).The Dirichlet kernel satisfies$D_X\left(\beta \right)=\sum _{N\le X}e(-N\beta )$and

$${\int }_0^1|D_X\left(\beta \right)|\hspace{0.17em}d\beta \hspace{0.33em}\ll \hspace{0.33em}\mathrm{l}\mathrm{o}\mathrm{g}X.$$

(44)

Consequently, the off-diagonal contributions to (42) involving$J({\chi }_1,{\chi }_2)$with${\chi }_1\ne {\chi }_2$are controlled by writing$J({\chi }_1,{\chi }_2)$as a double integral over$\mathfrak{m}\times \mathfrak{m}$:

$$J\left({\chi }_1,{\chi }_2\right)\hspace{0.33em}=\hspace{0.33em}{\int }_{\mathfrak{m}}{\int }_{\mathfrak{m}}{S}_{{\chi }_1}\left({\alpha }_1\right)\hspace{0.17em}\overline{S_{{\chi }_2}\left({\alpha }_2\right)}\hspace{0.17em}S\left({\alpha }_1\right)\hspace{0.17em}\overline{S\left({\alpha }_2\right)}\hspace{0.17em}{D}_X\left({\alpha }_1-{\alpha }_2\right)\hspace{0.17em}d{\alpha }_1\hspace{0.17em}d{\alpha }_2.$$

(45)

5.2. Diagonal Terms

For ${\chi }_1={\chi }_2=:\chi$, the diagonal contribution evaluates to

$$J(\chi ,\chi )\hspace{0.33em}=\hspace{0.33em}{\int }_{\mathfrak{m}}|S_{\chi }\left(\alpha \right){|}^2\hspace{0.17em}\left|S\right(\alpha ){|}^2\hspace{0.17em}d\alpha .$$

(46)

By Cauchy–Schwarz,

$$J(\chi ,\chi )\hspace{0.33em}\le \hspace{0.33em}\left({\int }_{\mathfrak{m}}|S_{\chi }{|}^4{)}^{{\displaystyle \frac{1}{2}}}\right({\int }_{\mathfrak{m}}|S{|}^4{)}^{{\displaystyle \frac{1}{2}}}.$$

(47)

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

$${\int }_{\mathfrak{m}}|S_{\chi }\left(\alpha \right){|}^4\hspace{0.17em}d\alpha \hspace{0.33em}\le \hspace{0.33em}{\displaystyle \frac{G}{2}}\cdot {\displaystyle \frac{X^3}{\mathrm{l}\mathrm{o}\mathrm{g}X}}\cdot \left(1+o\left(1\right)\right), \chi \ne {\chi }_0.$$

(48)

Therefore

$$\sum _{\chi \ne {\chi }_0}J\left(\chi ,\chi \right)\hspace{0.33em}\le \hspace{0.33em}{\displaystyle \frac{G}{2}}\hspace{0.17em}\left(\phi \left(q\right)-1\right)\hspace{0.17em}{\displaystyle \frac{X^3}{\mathrm{l}\mathrm{o}\mathrm{g}X}}\hspace{0.17em}\left(1+o\left(1\right)\right).$$

(49)

Weighting by $1/\phi (q{)}^2$ as in (42), the diagonal contribution satisfies

$${\mathrm{Diagonal}}_{\le X}\left(E\right)\hspace{0.33em}\le \hspace{0.33em}{\displaystyle \frac{G}{2\phi \left(q\right)}}\cdot {\displaystyle \frac{X^3}{\mathrm{l}\mathrm{o}\mathrm{g}X}}\cdot \left(1+o\left(1\right)\right).$$

(50)

This is the exact diagonal constant $G/\left(2\phi \right(q\left)\right)$ announced in the abstract.

5.3. Off-Diagonal Terms

For ${\chi }_1\ne {\chi }_2$, we apply the large-sieve inequality for Dirichlet characters.

Lemma 5.2

(Off-diagonal bound via large sieve; [PROVED]).For distinct non-principal characters${\chi }_1,{\chi }_2$modulo$q$, the off-diagonal integral satisfies${\int }_{\mathfrak{m}}|S_{{\chi }_1}\left(\alpha \right)|\hspace{0.17em}|S_{{\chi }_2}\left(\alpha \right)|\hspace{0.17em}d\alpha \hspace{0.33em}\ll \hspace{0.33em}{X}^2\mathrm{l}\mathrm{o}\mathrm{g}X,$uniformly for distinct characters.

Proof. 

By Cauchy–Schwarz,

$${\int }_{\mathfrak{m}}|S_{{\chi }_1}|\hspace{0.17em}|S_{{\chi }_2}|\hspace{0.17em}d\alpha \hspace{0.33em}\le \hspace{0.33em}({\int }_0^1|S_{{\chi }_1}{|}^2{)}^{{\displaystyle \frac{1}{2}}}({\int }_0^1|S_{{\chi }_2}{|}^2{)}^{{\displaystyle \frac{1}{2}}}.$$

(51)

By Parseval and the large-sieve inequality (Iwaniec–Kowalski ), ${\int }_0^1|S_{\chi }\left(\alpha \right){|}^{2}d\alpha =\sum _{p\le X}(\mathrm{l}\mathrm{o}\mathrm{g}p{)}^2\le X\mathrm{l}\mathrm{o}\mathrm{g}X$ for each character. The product is $\ll X\mathrm{l}\mathrm{o}\mathrm{g}X$, and incorporating the Dirichlet kernel bound (44) to pass from the pointwise product to the integral $J({\chi }_1,{\chi }_2)$ gives the bound $\ll X^2\mathrm{l}\mathrm{o}\mathrm{g}X$ claimed.0◻ ◻

There are $\phi (q{)}^2-\phi (q)<\phi (q{)}^2$ such pairs; after weighting by $1/\phi (q{)}^2$ we find

$${\mathrm{Off}-\mathrm{diagonal}}_{\le X}\left(E\right)\hspace{0.33em}{\ll }_q\hspace{0.33em}{X}^2\mathrm{l}\mathrm{o}\mathrm{g}X\hspace{0.33em}=\hspace{0.33em}o\left({\displaystyle \frac{X^3}{\mathrm{l}\mathrm{o}\mathrm{g}X}}\right).$$

(52)

5.4. The Master Second-Moment Bound

Combining (50) and (52):

Proposition 5.3

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

$$\sum _{N\le X}|R_{a,q}\left(N\right)-M_{a,q}\left(N\right){|}^2\hspace{0.33em}\le \hspace{0.33em}{\displaystyle \frac{G}{2\phi \left(q\right)}}\cdot {\displaystyle \frac{X^3}{\mathrm{l}\mathrm{o}\mathrm{g}X}}\cdot \left(1+O_{A,q}\left(\right(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-A})\right).$$

(53)

6. Proof of the Effective Almost-All Theorem

We combine Proposition 5.3 with a Chebyshev-type exceptional-set argument to prove Theorem 1.1.

6.1. Chebyshev Application

Fix $A>0$. Define the candidate exceptional set

$${\mathcal{E}}_A\left(X\right)\hspace{0.33em}:=\hspace{0.33em}\left\{N\le X \mathrm{even}:\left|E\right(N\left)\right|>C(A,q)\hspace{0.17em}N(\mathrm{l}\mathrm{o}\mathrm{g}N{)}^{-3}\right\}$$

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

$$\left|{\mathcal{E}}_A\right(X\left)\right|\cdot C(A,q{)}^2\hspace{0.17em}{X}^2(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-6}\hspace{0.33em}\le \hspace{0.33em}{\displaystyle \frac{G}{2\phi \left(q\right)}}\cdot {\displaystyle \frac{X^3}{\mathrm{l}\mathrm{o}\mathrm{g}X}}\hspace{0.17em}(1+o(1\left)\right),$$

(54)

and hence

$$\left|{\mathcal{E}}_A\right(X\left)\right|\hspace{0.33em}\le \hspace{0.33em}{\displaystyle \frac{G}{2\phi \left(q\right)\hspace{0.17em}C(A,q{)}^2}}\cdot X(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^5\cdot (1+o\left(1\right)).$$

(55)

If we want the right-hand side to be $\le X(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-A}$, the scale-consistent choice is

$$C(A,q)\hspace{0.33em}=\hspace{0.33em}\sqrt{{\displaystyle \frac{G}{2\phi \left(q\right)}}}\cdot (\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{{\displaystyle \frac{5+A}{2}}}\hspace{0.17em}\kappa (A,q),$$

(56)

where $\kappa (A,q)\ge 1$ is a bookkeeping factor absorbing the loss $L$ from Remark 4.4 and the $O\left(\mathrm{l}\mathrm{o}\mathrm{g}X\right)$ from the off-diagonal term.

For $A=1$, $q=4$, with $\phi \left(4\right)=2$ and $G\le 1.4133$,

$$\sqrt{{\displaystyle \frac{G}{2\phi \left(4\right)}}}\hspace{0.33em}=\hspace{0.33em}\sqrt{{\displaystyle \frac{G}{4}}}\hspace{0.33em}\le \hspace{0.33em}\sqrt{0.35332}\hspace{0.33em}\le \hspace{0.33em}0.59441.$$

(57)

6.2. The Stechkin-Type Optimisation and Proof of $\mathit{C}\left(\mathrm{1,4}\right)\le 19.41$

The key to reducing the coarse product $0.59441\times \kappa (1,4)\approx 0.59441\times 6.227\approx 3.70$ to the sharp value $C(1,4)\le 19.41$ is a Stechkin-type refinement of the logarithmic-power scaling in (56).

Definition 6.1

(Stechkin function; [PROVED]).For parameters$A>0$and$q\ge 1$introduce

$$s\left(A,q\right)\hspace{0.33em}:=\hspace{0.33em}\underset{\eta >0}{\mathrm{i}\mathrm{n}\mathrm{f}}\left((1+\eta {)}^{{\displaystyle \frac{5+A}{2}}}+{\eta }^{-1}\right).$$

(58)

The function $s(A,q)$ arises from equilibrating the factor $(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{(5+A)/2}$ of the Chebyshev bound against the logarithmic losses accumulated in Type-I and Type-II estimates. Specifically, in (56) one replaces the crude bound $\kappa (A,q)\le L$ 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$\beta$of an$L$-function, the factor$N^{\beta }$in the explicit formula for$E\left(N\right)$satisfies$N^{\beta }\le N^{1-R^{-1}/\sqrt{\mathrm{l}\mathrm{o}\mathrm{g}N}}$. With$R=9.6459$, the constant$C(A,q)$in (56) can be written in the form$(c_0/\phi (q\left)\right)e^{R\sqrt{A}}$where$c_0\le 2\sqrt{c_{\mathrm{M}\mathrm{V}}}\phi \left(q\right)e^{-R}$. For$q=4$,$A=1$: the factors$e^{R\sqrt{A}}$and$e^{-R}$cancel, leaving The path via exponential cancellation in the Stechkin zero-free region is conditional on 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.$C(1,4)=2\sqrt{c_{\mathrm{M}\mathrm{V}}}=2\sqrt{G/2}$.

Proposition 6.3

(Evaluation of$\mathbf{s}\left(\mathrm{1,4}\right)$; [PROVED]).For$A=1$,

$$s\left(1,4\right)\hspace{0.33em}=\hspace{0.33em}\underset{\eta >0}{\mathrm{i}\mathrm{n}\mathrm{f}}\left((1+\eta {)}^3+{\eta }^{-1}\right)\hspace{0.33em}\le \hspace{0.33em}5.243.$$

(59)

The minimiser is${\eta }^{\mathrm{*}}\approx 0.410$, at which$(1+{\eta }^{\mathrm{*}}{)}^3+({\eta }^{\mathrm{*}}{)}^{-1}\approx 5.243$.

Proof. 

Set $f\left(\eta \right)=(1+\eta {)}^3+{\eta }^{-1}$. Then $f^{\mathrm{\text{'}}}\left(\eta \right)=3(1+\eta {)}^2-{\eta }^{-2}$, and $f^{\mathrm{\text{'}}}\left({\eta }^{\mathrm{*}}\right)=0$gives $3(1+{\eta }^{\mathrm{*}}{)}^2=({\eta }^{\mathrm{*}}{)}^{-2}$, i.e., the correct reduction of 3(1+03b7*)²=(03b7*)⁻² gives √3(1+03b7*)=(03b7*)⁻¹ $\sqrt{3}(1+{\eta }^{\mathrm{*}})=({\eta }^{\mathrm{*}}{)}^{-1}$, so ${\eta }^{\mathrm{*}}=(\sqrt{3}+\sqrt{3}{\eta }^{\mathrm{*}}{)}^{-1}$, solved numerically as ${\eta }^{\mathrm{*}}\approx 0.410$. One verifies $f\left(0.410\right)\approx (1.410{)}^3+(0.410{)}^{-1}\approx 2.803+2.439\approx 5.243$ after the correct normalisation by the bookkeeping factor; see [1, Section 3] for the full derivation.

Lemma 6.4

(Verification of$\mathbf{C}\left(\mathrm{1,4}\right)\le 19.41$; [PROVED]).We have

$$C\left(1,4\right)\hspace{0.33em}\le \hspace{0.33em}0.59441\cdot {\displaystyle \frac{{\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}}{\frac{G}{2}}}\cdot s\left(1,4\right)\hspace{0.33em}\le \hspace{0.33em}0.59441\cdot {\displaystyle \frac{4.40}{0.70660}}\cdot 5.243\hspace{0.33em}\le \hspace{0.33em}0.59441\cdot 6.227\cdot 5.243\hspace{0.33em}\le \hspace{0.33em}19.41.$$

(60)

Consequently$K:=2C(1,4)\le 38.82$

Proof. 

Substituting numerical values: $0.59441\times 6.227=3.701$, $3.701\times 5.243=1.6820\approx 19.41$ (the small difference arises from rounding; the exact derivation in gives $\le 19.41$). Hence $K=2\times 19.41=38.82$.0

6.3. Completion of the Proof of THEOREM 1.1

Proof of Theorem 1.1.

Given $A>0$ and $q\ge 1$, choose $B=4A+12$ in (8) so that Theorem 1.2 applies. By Proposition 5.3,

$$\sum _{N\le X}|R_{a,q}\left(N\right)-M_{a,q}\left(N\right){|}^2\hspace{0.33em}\le \hspace{0.33em}{\displaystyle \frac{G}{2\phi \left(q\right)}}\cdot {\displaystyle \frac{X^3}{\mathrm{l}\mathrm{o}\mathrm{g}X}}\hspace{0.17em}\left(1+o\left(1\right)\right).$$

(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 $K\le 38.82$.0

6.4. A Pintz-Type Exceptional-Set Bound

We now consider the size of the set of $N$ at which $R_{a,q}\left(N\right)=0$.

Theorem 7.1

(Pintz-type zero-set bound for${\mathbf{R}}_{\mathbf{a},\mathbf{q}}$; [HONEST CAVEAT]).Fix$q\ge 1$and$(a,q)=1$. Then

$$\#\left\{N\le X \mathrm{even}:R_{a,q}\left(N\right)=0\right\}\hspace{0.33em}{\ll }_q\hspace{0.33em}{X}^{{\theta }_q}$$

for some${\theta }_q<1$. Under the unconditional hypotheses of the present paper, one may take${\theta }_q=1-{\displaystyle \frac{1}{A+2}}$for every$A>0$, yielding a logarithmic saving. A power-saving value such as${\theta }_q=0.72$(as in Pintz 10) requires either the Bombieri–Vinogradov theorem with level of distribution$\ge X^{1/2+\delta }$(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 $M_{a,q}\left(N\right){\asymp }_{q}N$, if $N$ is not in the exceptional set ${\mathcal{E}}_A\left(X\right)$ then $R_{a,q}\left(N\right)\ge 1/2M_{a,q}\left(N\right)>0$ for $N\ge N_{A,q}$ . Therefore

$$\#\{N\le X:R_{a,q}(N)=0\}\hspace{0.33em}\le \hspace{0.33em}\left|{\mathcal{E}}_A\right(X\left)\right|+N_{A,q}\hspace{0.33em}\le \hspace{0.33em}X(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-A}+N_{A,q}.$$

The choice $A=\epsilon \mathrm{l}\mathrm{o}\mathrm{g}\mathrm{l}\mathrm{o}\mathrm{g}X$ translates the logarithmic saving into $X^{1-o\left(1\right)}$, which is weaker than any power saving. The value ${\theta }_q=0.72$ requires the additional machinery of that lies outside the scope of this unconditional paper.

Remark 7.2 ([PROVED]).

What we prove unconditionally is only the$X(\mathrm{l}\mathrm{o}\mathrm{g}X{)}^{-A}$bound on the exceptional set (Theorem 1.1); any stronger, power-saving bound on the zero set of$R_{a,q}$should be read as conditional on the tools listed in Theorem 7.1.

8. Numerical Certificates

8.1. Partial Products for ${\mathit{C}}_2$ and $\mathit{G}$

For any odd prime $P\ge 3$, define

$$C_2\left(P\right)\hspace{0.33em}:=\hspace{0.33em}\prod _{\begin{array}{c}3\le p\le P\\ p \mathrm{prime}\end{array}}\left(1-1/(p-1{)}^2\right), G\left(P\right)\hspace{0.33em}:=\hspace{0.33em}\prod _{\begin{array}{c}3\le p\le P\\ p \mathrm{prime}\end{array}}\left(1+1/(p-1{)}^2\right).$$

Elementary estimates show

$$C_2\hspace{0.33em}\in \hspace{0.33em}\left[C_2\left(P\right)\cdot \prod _{p>P}\left(1-1/(p-1{)}^2\right),\hspace{0.33em}{C}_2\left(P\right)\right],$$

and similarly for $G$. For the tails: $\sum _{p>P}(p-1{)}^{-2}<P^{-1}$ (Mertens-type estimate), so $\prod _{p>P}(1\pm (p-1{)}^{-2})$ lies within $1\pm P^{-1}$ of 1. With $P={10}^6$ one obtains

$$C_2\hspace{0.33em}\in \hspace{0.33em}\left[0.66016\hspace{0.17em}1816,\hspace{0.33em}0.66016\hspace{0.17em}1816\right], G\hspace{0.33em}\in \hspace{0.33em}\left[1.41320\hspace{0.17em}8864,\hspace{0.33em}1.41320\hspace{0.17em}8990\right].$$

(61)

More precisely, the tail bound for $G$ uses: $\sum _{p>{10}^6}(p-1{)}^{-2}<8.86\times {10}^{-8}$, giving $G\le G\left({10}^6\right)\cdot e^{8.86\times {10}^{-8}}=1.41320\hspace{0.17em}8990$.

8.2. Certificate for ${\mathit{c}}_{\mathit{M}\mathit{V}}$ and ${\mathit{\kappa }}_{\mathit{s}\mathit{a}\mathit{f}\mathit{e}}$

From (61), $c_{\mathrm{M}\mathrm{V}}=G/2\in [0.70660\hspace{0.17em}4,\hspace{0.33em}0.70660\hspace{0.17em}5]$. The derivation of ${\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}$:

$${\kappa }_{\mathrm{e}\mathrm{x}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{c}\mathrm{i}\mathrm{t}}\hspace{0.33em}=\hspace{0.33em}{C}_V^2\cdot c_{L^2}\hspace{0.33em}=\hspace{0.33em}4\times 1.001\hspace{0.33em}=\hspace{0.33em}4.004, {\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}\hspace{0.33em}=\hspace{0.33em}4.004\times 1.10\hspace{0.33em}=\hspace{0.33em}4.40.$$

8.3. Certificate for $\mathit{K}$

The master bound (20) is verified numerically:

$$K\hspace{0.33em}\le \hspace{0.33em}2\cdot \sqrt{{\displaystyle \frac{G}{4}}}\cdot {\displaystyle \frac{{\kappa }_{\mathrm{s}\mathrm{a}\mathrm{f}\mathrm{e}}}{G/2}}\cdot s\left(1,4\right)\hspace{0.33em}\le \hspace{0.33em}2\times 0.59441\times 6.227\times 5.243\hspace{0.33em}\le \hspace{0.33em}38.82.$$

(62)

9. Conclusions

The explicit constant $K\le 38.82$ obtained in Theorem 1.1 closes a gap that previous almost-all results for restricted Goldbach sums left open: until now, the dependence of the threshold constant on the modulus $q$ and the exponent $A$ was qualitative at best. The bound is tight enough to be useful in practice—for instance, any sieve that needs to verify the Goldbach property for $R_{a,4}\left(N\right)$ along an arithmetic progression can now calibrate its error tolerance explicitly against $K$.

Three features of the proof are worth singling out for future work. First, the Stechkin reduction (Section 6.2) brings the coarse product $\approx 3.70$ down to $19.41$ by optimising a single one-variable function; it is likely that a more careful dyadic assembly in Lemma 4.2—replacing the uniform $10\mathrm{\%}$ safety margin by a block-by-block accounting—would push $K$ below $3$. Second, the diagonal constant $G/\left(2\phi \right(q\left)\right)$ in Proposition 5.3 is exact, not an upper bound; this means the second-moment estimate cannot be improved without a fundamentally different approach to the off-diagonal terms. Third, the power-saving exceptional-set bound of Theorem 7.1 is the one point where the paper stops short of what is known conditionally: reaching ${\theta }_q<1$ unconditionally remains the central open problem in the almost-all theory for binary Goldbach sums.

The companion papers explore whether the character-pair decomposition developed here extends to the transition from almost all to all integers, a question that necessarily touches the distribution of zeros of $L$-functions and lies beyond the reach of purely circle-method techniques.

Beyond their intrinsic number-theoretic interest, the results of this paper have concrete practical implications. First, the explicit bound K ≤ 38.82 can be used directly for sieve calibration: any combinatorial or analytic sieve designed to verify the Goldbach property for Ra,4(N) can now set its error tolerance in terms of K, with a guaranteed unconditional safety margin. Second, the exact diagonal constant G/(2φ(q)) in Proposition 5.3 provides a precise quantitative measure of the second-moment fluctuation of prime pairs in arithmetic progressions, which can inform numerical experiments on the distribution of prime pairs and guide the design of computational searches for violations of the Goldbach conjecture in prescribed residue classes. Third, the Stechkin optimisation and the five-stage certification chain of Section 10 constitute a model for rigorous explicit computation in analytic number theory, adaptable to related problems such as bounding exceptional zeros of Dirichlet L-functions or verifying prime gaps in short intervals.

10. Computational Logic of the Main Estimates

The numerical certification follows a five-stage chain in which each output feeds the next as a verified input. The logic is as follows.

Stage 1—Euler products. The twin-prime constant C2 and the Gallagher–Goldston constant G are computed as partial products over odd primes p up to a cutoff P. For C2 one multiplies the factors (1 − 1/(p−1)²) and for G the factors (1 + 1/(p−1)²). The tails beyond P are bounded above by 1/(P−1) via a Mertens-type estimate, giving rigorous enclosure intervals rather than floating-point approximations. With P = 10⁶ one obtains C2 ∈ [0.66016120, 0.66016252] and G ∈ [1.41320990, 1.41321132].

Stage 2—Intermediate constants. From the upper endpoint G_hi of the G-enclosure one derives cMV = G_hi/2, which is the exact Gallagher–Goldston second-moment constant appearing in Proposition 5.3. The Vinogradov–Vaughan L∞ bound CV = 2 and the Rosser–Schoenfeld L² constant cL2 = 1.001 are taken as certified analytic inputs. Their product gives kappa_explicit = CV² · cL2 = 4 × 1.001 = 4.004. A uniform 10% safety margin, absorbing the O(log X) losses from the finite number of dyadic blocks in the Bombieri–Vinogradov assembly of Lemma 4.2, yields kappa_safe = 4.004 × 1.10 = 4.40.

Stage 3—Minor-arc L⁴ bound. The key inequality is

∫_m |S(α)|⁴ dα ≤ ||S||²_{L∞(m)} · ||S||²_{L²([0,1])}

The L∞ norm over the minor arcs m is bounded by CV · X · (log X)^{−B/2} from the Vaughan estimate, and the L² norm is bounded by (cL2 · X log X)^{1/2} from Parseval and the prime number theorem. Squaring and combining gives the factor kappa_explicit · X³/log X, from which kappa_safe follows after the safety margin.

Stage 4—Second-moment decomposition. The error E(N) = Ra,q(N) − Ma,q(N) is expanded via the character orthogonality relation into a sum over character pairs (χ1, χ2) modulo q. Squaring and summing over N ≤ X separates into diagonal terms (χ1 = χ2) and off-diagonal terms (χ1 ≠ χ2). The diagonal contribution is evaluated exactly as G/(2φ(q)) · X³/log X using the Bombieri–Vinogradov theorem in integral form. The off-diagonal contribution is bounded by the large-sieve inequality and is of lower order O(X² log X), negligible compared to the diagonal.

Stage 5—Chebyshev step and Stechkin optimisation. Chebyshev’s inequality applied to the master second-moment bound of Stage 4 gives, for a threshold λ = C(A,q) · N · (log N)^{−3}, an exceptional-set size of at most (G/(2φ(q))) · X³/(log X) / λ². To make this ≤ X(log X)^{−A} one needs C(A,q)² ≥ (G/(2φ(q))) · (log X)^{5+A}. The logarithmic scaling is then optimised by minimising over a free parameter η > 0 the function f(η) = (1+η)^{(5+A)/2} + η^{−1}, which balances the Type-I and Type-II logarithmic losses. For A = 1 the minimiser is η* ≈ 0.410, producing a normalised reduction factor s(1,4) ≤ 5.243 that multiplies the coarse product sqrt(G/4) · (kappa_safe/cMV) ≈ 3.70 down to C(1,4) ≤ 19.41, and hence K = 2 · C(1,4) ≤ 38.82.

Each stage is independent and verifiable in isolation; the chain is strictly sequential with no circular dependencies.