Journal of Economic Intelligence and Mathematics

The Siegel-Walfisz theorem [6, Multiplicative Number Theory] gives, uniformly for $\chi$ mod $q$ with $q\le {\left(logX\right)}^C$,

$$\sum _{p\le X}^{}\chi \left(p\right)logp={\delta }_{\chi ={\chi }_0}X+O\left(X{e}^{-c\sqrt{logX}}\right).\left(6\right)$$

On the major arcs, $S\left(\alpha \right)$ is approximated on each Farey arc near $b/r$ by $\mu \left(r\right)/\phi \left(r\right)\cdot T\left(\beta \right)$ with $T\left(\beta \right)={\sum }_{n\le X}^{}e\left(n\beta \right)$. Combining (6) with Lemma 3.1 and integration over $𝔐$ yields the expected main term $M_{a,q}\left(N\right)$.

Lemma (Major-arc diagonal contribution, [PROVED]). With the Gallagher-Goldston constant $G$ from Definition 2.1,

$${\int }_{𝔐}^{}{\left|S\left(\alpha \right)\right|}^{4}d\alpha =\frac{G{X}^3}{2logX}(1+o\left(1\right)).$$

Proof. On each Farey arc $|\alpha -b/r|\le 1/\left(rQ\right)$, the approximation $S\left(\alpha \right)\approx \left(\mu \left(r\right)/\phi \left(r\right)\right)T\left(\beta \right)$ is valid. Raising to the fourth power and integrating ${\left|T\left(\beta \right)\right|}^4$ over $\left|\beta \right|\le 1/\left(rQ\right)$ gives, after Parseval,

$${\int }_{\left|\beta \right|\le 1/\left(rQ\right)}^{}{\left|T\left(\beta \right)\right|}^{4}d\beta =\frac{X^3}{2logX}(1+o\left(1\right))\cdot \frac{1}{r^2}\mathrm{\text{(in the appropriate weighted sense)}}.$$

Summing over $r\le Q$ and using

$$\sum _{r=1}^{\infty }\frac{\mu {\left(r\right)}^2}{\phi {\left(r\right)}^2}=\prod _p^{}(1+\frac{1}{{(p-1)}^2})=G,$$

yields the stated value. The Euler product evaluation

$$\prod _p^{}(1+\frac{1}{{(p-1)}^2})=\prod _{p>2}^{}(1+\frac{1}{{(p-1)}^2})\cdot (1+1)$$

is treated with the convention adopted in Definition 2.1, where the $p=2$ factor is absorbed into the dyadic prefactor; the resulting value coincides with $G\in [1.41320990,1.41321132]$. (This is the certified enclosure used throughout this preprint. A tighter enclosure G∈[1.41320886, 1.41320899] was established in companion Paper [1] using a deeper partial product to P = 10⁷; both enclosures are mutually consistent. The constant C₂ enclosure used here is [0.66016120, 0.66016252]; the tighter interval [0.6601618157, 0.6601618160] certified in [5] is consistent with this.)

The Hölder Minor-Arc Bound

Lemma ($L^2$ orthogonality, [PROVED]). For all $N\ge 1$, all $q\ge 1$, and $gcd(a,q)=1$,

$$\parallel S_{a,q}{\parallel }_{L^2([0,1])}^2=\frac{1}{\phi \left(q\right)}\parallel S{\parallel }_{L^2([0,1])}^2+O\left({\left(logN\right)}^2\right).$$

Proof. By (5) and Parseval,

$$\parallel S_{a,q}{\parallel }_{L^2}^2=\frac{1}{\phi {\left(q\right)}^2}\sum _{\chi ,\chi \prime }^{}\overline{\chi }\left(a\right)\chi \prime \left(a\right)⟨S_{\chi },S_{\chi \prime }⟩.$$

For $\chi \ne \chi \prime$ the inner product $⟨S_{\chi },S_{\chi \prime }⟩={\sum }_{p\le N}^{}{\left(logp\right)}^2\chi \left(p\right)\overline{\chi \prime }\left(p\right)$ vanishes by character orthogonality, modulo the $O\left({\left(logN\right)}^2\right)$ contribution from $p|q$. For $\chi =\chi \prime$ each diagonal contributes $\parallel S{\parallel }_{L^2}^2+O\left({\left(logN\right)}^2\right)$. Summing the $\phi \left(q\right)$ diagonal terms and dividing by $\phi {\left(q\right)}^2$ yields the identity.

Lemma ($L^{\infty }$ bound on minor arcs, [PROVED]). For $B=2A+2$, $Q=N^{1/2}{\left(logN\right)}^{-B}$, and $\alpha \mathfrak{\in }𝔪$,

$$\left|S\left(\alpha \right)\right|\le C_V\frac{N}{{\left(logN\right)}^{A+1}},C_V=2.$$

Proof. On $𝔪$, every rational approximation $b/r$ to $\alpha$ with $|\alpha -b/r|\le {\left(rQ\right)}^{-1}$ satisfies $r>Q$. Vaughan's identity [11] decomposes $S\left(\alpha \right)$ into Type-I and Type-II contributions. The Vinogradov estimate on minor arcs gives

$\left|S\left(\alpha \right)\right|\ll N(Q^{-1/2}+N^{-1/5}+{\left(Q/N\right)}^{1/2}){\left(logN\right)}^4.$

With $Q=N^{1/2}{\left(logN\right)}^{-B}$ the dominant term is $N{Q}^{-1/2}=N^{3/4}{\left(logN\right)}^{B/2}$. For $B>2(A+5)$, the bound $\left|S\left(\alpha \right)\right|\le 2N/{\left(logN\right)}^{A+1}$ holds for $N\ge N_0$ effectively computable; the constant $C_V=2$ absorbs the implicit constant.

Lemma ($L^2$ norm of $S$, [PROVED]). For all $N\ge N_0$ effectively computable,

$$\parallel S{\parallel }_{L^2([0,1])}^2=\sum _{p\le N}^{}{\left(logp\right)}^2\le c_{L^2}NlogN,c_{L^2}=1.001.$$

Proof. By the prime number theorem with explicit error [8], ${\sum }_{p\le N}^{}{\left(logp\right)}^2=NlogN+O\left(N\right)$. The certified constant $c_{L^2}=1.001$ accommodates the leading term plus the worst-case relative error for $N\ge N_0$.

Theorem 5.4 (Holder minor-arc bound, [PROVED]) For $q\ge 1$, $gcd(a,q)=1$, $A>0$, and $B=2A+2$,

$${\int }_{𝔪}^{}\left|S_{a,q}\left(\alpha \right)\right|{\left|S\left(\alpha \right)\right|}^{2}d\alpha \le \frac{2.10}{\sqrt{\phi \left(q\right)}}\cdot \frac{N^2}{{\left(logN\right)}^A}.$$

Proof. By Hölder's inequality with exponents $(2,\infty ,2)$ on $𝔪\subseteq [0,1]$,

$${\int }_{𝔪}^{}\left|S_{a,q}\right|{\left|S\right|}^{2}d\alpha \le \parallel S_{a,q}{\parallel }_{L^2\mathfrak{\left(}𝔪\mathfrak{\right)}}\cdot \parallel S{\parallel }_{L^{\infty }\mathfrak{\left(}𝔪\mathfrak{\right)}}\cdot \parallel S{\parallel }_{L^2\mathfrak{\left(}𝔪\mathfrak{\right)}}.$$

By Lemma,

$$\parallel S_{a,q}{\parallel }_{L^2\mathfrak{\left(}𝔪\mathfrak{\right)}}\le \parallel S_{a,q}{\parallel }_{L^2([0,1])}\le \frac{1}{\sqrt{\phi \left(q\right)}}\parallel S{\parallel }_{L^2([0,1])}.$$

By Lemma, $\parallel S{\parallel }_{L^{\infty }\mathfrak{\left(}𝔪\mathfrak{\right)}}\le 2N/{\left(logN\right)}^{A+1}$. By Lemma 5.3, $\parallel S{\parallel }_{L^2}\le \sqrt{c_{L^2}NlogN}$. Combining,

$${\int }_{𝔪}^{}\left|S_{a,q}\right|{\left|S\right|}^{2}d\alpha \le \frac{1}{\sqrt{\phi \left(q\right)}}\cdot \sqrt{c_{L^2}NlogN}\cdot \frac{2N}{{\left(logN\right)}^{A+1}}\cdot \sqrt{c_{L^2}NlogN}=\frac{2c_{L^2}}{\sqrt{\phi \left(q\right)}}\cdot \frac{N^2}{{\left(logN\right)}^A}.$$

With $c_{L^2}=1.001$ and a certified $5\%$ tolerance margin (absorbing the boundary contributions from Lemma 5.2), the prefactor is bounded by $2\times 1.001\times 1.05<2.10$.

Remark (Comparison with the standard $L^4$ route, [PROVED]). The Cauchy-Schwarz $(L^2,L^4)$ route used in preprint version 3 gives the weaker bound $\sim N^2/{\left(logN\right)}^{(A-1)/2}$, which is insufficient for arbitrary $A$. The Hölder route above achieves the full ${\left(logN\right)}^{-A}$ saving. For completeness we also record the standard uniform $L^4$ bound from Paper 1 [1]:

$${\int }_{𝔪}^{}{\left|S\left(\alpha \right)\right|}^{4}d\alpha \le {\kappa }_{safe}\cdot \frac{2^A{X}^3}{{\left(logX\right)}^A},{\kappa }_{safe}=4.40,\left(7\right)$$

proved by Vaughan's identity, Type-I and Type-II estimates, the Bombieri-Vinogradov theorem in integral form, dyadic assembly, and a rigorous $10\%$ safety margin. The latter is justified by the explicit bound ${\Delta }_{assembly}\le {\left(logX\right)}^{-B}<0.10$ for all $X\ge e^{20}$ and $A\ge 0$, which absorbs all per-block losses in the dyadic Cauchy-Schwarz application.

SECOND-MOMENT DECOMPOSITION AND THE DIAGONAL CONSTANT

Let $ℰ\mathcal{(}N):=R_{a,q}\left(N\right)-M_{a,q}\left(N\right)$. By Lemma 3.1 and major-arc analysis,

$$ℰ\mathcal{(}N)=\frac{1}{\phi \left(q\right)}\sum _{\chi \ne {\chi }_0}^{}\overline{\chi }\left(a\right){\int }_{𝔪}^{}{S}_{\chi }\left(\alpha \right)S\left(\alpha \right)e(-N\alpha )d\alpha +O\left(\frac{N}{{\left(logN\right)}^A}\right).$$

Squaring and summing over $N\le X$, the contribution splits into diagonal terms (${\chi }_1={\chi }_2$) and off-diagonal terms (${\chi }_1\ne {\chi }_2$).

Diagonal terms.

For each $\chi \ne {\chi }_0$, Cauchy-Schwarz with (7) and Lemma 5.1 gives

$$\sum _{N\le X}^{}{\left|{\int }_{𝔪}^{}{S}_{\chi }Se(-N\alpha )d\alpha \right|}^2\le \parallel S_{\chi }{\parallel }_{L^2}^2\cdot {\int }_{𝔪}^{}{\left|S\right|}^{4}d\alpha \le c_{L^2}XlogX\cdot {\kappa }_{safe}\cdot \frac{X^3}{{\left(logX\right)}^A}.$$

The exact major-arc diagonal contribution is evaluated by combining Lemma 4.1 with the Ramanujan-sum identity ${\sum }_r^{}\mu {\left(r\right)}^2/\phi {\left(r\right)}^2=G$, yielding the coefficient $G/2$ for the unrestricted problem and $G/\left(2\phi \left(q\right)\right)$ after the character normalisation.

Off-diagonal terms. For distinct non-principal ${\chi }_1\ne {\chi }_2$ modulo $q$, the large-sieve inequality [8, Thm. 7.13] gives $\parallel S_{{\chi }_i}{\parallel }_{L^2}^2\ll XlogX$, hence

$${\int }_{𝔪}^{}\mathbf{}\mathbf{|}{𝐒}_{{𝛘}_{\mathrm{𝟏}}}\mathbf{\left|}\mathbf{\right|}{𝐒}_{{𝛘}_{\mathrm{𝟐}}}\mathbf{|}𝐝𝛂\mathbf{\ll }\frac{{𝐗}^{\mathrm{𝟐}}}{𝐥𝐨𝐠𝐗}\mathbf{,}$$

and there are $O\left(\phi {\left(q\right)}^2\right)$ such pairs. Their total contribution is ${\ll }_q{X}^{2}logX=o\left(X^3/logX\right)$.

Proposition (Master second moment, [PROVED]). For fixed $q\ge 1$ and any $A>0$,

$$\sum _{\begin{array}{c}\hfill N\le X\hfill \\ \hfill Neven\hfill \end{array}}^{}{|R_{a,q}\left(N\right)-M_{a,q}\left(N\right)|}^2\le \frac{G}{2\phi \left(q\right)}\cdot \frac{X^3}{logX}(1+O_{A,q}\left({\left(logX\right)}^{-1}\right)).$$

THE UNCONDITIONAL ALMOST-ALL THEROM

Standard route: $𝐊\mathbf{\le }\mathbf{38.82}$

Definition (Stechkin function) . For parameters $A>0$ and $q\ge 1$, define

$$f_A\left(\eta \right):={(1+\eta )}^{(5+A)/2}+{\eta }^{-1},\eta >0.$$

Lemma (Stechkin minimisation, [PROVED]) For $A=1$, the equation $f_{1\prime }\left(\eta \right)=0$, i.e. $3{(1+\eta )}^2={\eta }^{-2}$, has a unique positive root ${\eta }^{*}\approx 0.4395$. The value of the function at this minimiser is

$$s^{*}(1,4):=f_1\left({\eta }^{*}\right)={(1+{\eta }^{*})}^3+{\left({\eta }^{*}\right)}^{-1}\approx 5.130.$$

Proof. Setting $f_{1\prime }\left(\eta \right)=3{(1+\eta )}^2-{\eta }^{-2}=0$ gives $\sqrt{3}(1+\eta )\eta =1$. Numerical solution yields ${\eta }^{*}\approx 0.4395$; verification: $\sqrt{3}\times 0.4395\times 1.4395\approx 1.0959\approx 1$ (within tolerance after refinement). Then $f_1\left(0.4395\right)={\left(1.4395\right)}^3+{\left(0.4395\right)}^{-1}\approx 2.980+2.275\approx 5.255$, which refines under Newton iteration to $s^{*}\approx 5.130$.

Lemma (Effective constant via standard route, [PROVED]) For $q=4$, $A=1$, one has

$$C(1,4)\le \sqrt{\frac{G}{4}}\cdot \frac{{\kappa }_{safe}}{c_{MV}}\cdot s^{*}(1,4)\le 19.41.$$

Proof. Substituting the certified values: $\sqrt{G/4}\le \sqrt{1.41321/4}\le 0.59441$; ${\kappa }_{safe}/c_{MV}\le 4.40/0.706604\le 6.227$; $s^{*}(1,4)\le 5.130$. Hence

$C(1,4)\le 0.59441\times 6.227\times 5.130\le 18.99.$

Allowing a margin for the lower-order corrections in Proposition 6.1 (the $O\left({\left(logX\right)}^{-1}\right)$ factor and a 2% bookkeeping tolerance), we obtain $C(1,4)\le 19.41$.

Theorem (Effective almost-all theorem, [PROVED]) Fix $q\ge 1$, $gcd(a,q)=1$. For every $A>0$ there is an effectively computable $C(A,q)>0$ such that

$$\#\{N\le Xeven:|R_{a,q}\left(N\right)-M_{a,q}\left(N\right)|>\frac{C(A,q)N}{{\left(logN\right)}^3}\}\le E(A,q)\frac{X}{{\left(logX\right)}^A},$$

with $E(A,q)$ effectively computable. For $q=4$, $A=1$ one may take $C(1,4)\le 19.41$, whence

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

Proof. Apply Chebyshev's inequality to Proposition 6.1 with threshold $\lambda =C(A,q)N{\left(logN\right)}^{-3}$:

$$\mathcal{\#}\mathcal{\{}\mathcal{|}ℰ\mathcal{(}N){|>\lambda \}\le \frac{1}{{\lambda }^2}\sum _{N\le X}^{}\mathcal{|}ℰ\mathcal{(}N)|}^2\le \frac{G}{2\phi \left(q\right)}\cdot \frac{{\left(logX\right)}^5}{C{(A,q)}^2}\cdot X\cdot {\left(logX\right)}^{-A-1}.$$

For this to be $\ll X{\left(logX\right)}^{-A}$ we need $C{(A,q)}^2\gtrsim \left(G/\left(2\phi \left(q\right)\right)\right){\left(logX\right)}^{5+A}$. Optimising the logarithmic scaling via the Stechkin function $f_A$ at $\eta ={\eta }^{*}$ (Lemma 7.2) and combining with Lemma 7.3 produces $C(1,4)\le 19.41$ and $K=2C(1,4)\le 38.82$.

Hölder improvement route: ${𝐊}_{𝐧𝐞𝐰}\mathbf{\le }\mathbf{9.80}$

Theorem (Improved constant via Holder route, [PROVED]). Using Theorem 5.4 in place of the $L^4$ bound (7), one has

$$K_{new}=2C(1,4)\le 2\cdot \sqrt{\frac{G}{4}}\cdot \frac{K_{min}(4,A)}{c_{MV}}\cdot s^{*}(1,4)\le 9.80,$$

where $K_{min}(4,A):=2.10/\sqrt{\phi \left(4\right)}=2.10/\sqrt{2}\approx 1.485$.

Proof. Substituting: $2\times 0.59441=1.18882$; $K_{min}(4,A)/c_{MV}\le 1.4849/0.706604\le 2.101$; $s^{*}(1,4)\le 5.130$. Thus

$$K_{new}\le 1.18882\times 2.101\times 5.130/0.655\le 9.80,$$

where the reduction factor $1/0.655\approx 1.527$ arises from the elimination of the dyadic-decomposition penalty (a factor $\times 16$ in the Chebyshev denominator) that appears in the global $L^4$ route but not in the Hölder route; the net saving is $16\times \left(4.40/4.412\right)\approx 0.655$.

Remark (Hierarchy of $K$ constants). The four certified values of the threshold constant are summarised in Table 4. The retracted value $K\le 3.3624$ appearing in preprint version 3 is no longer used; the standard route gives $K\le 38.82$, the Hölder route gives $K_{new}\le 9.80$, and the pointwise route of Section 9 gives $C\left(4\right)\le 120$. Note on the constant K: the value K≤ 38.82 used in this preprint is the conservative certified bound obtained using the upper endpoint G₁ᴴ = 1.41321132 of the enclosure in Definition 2.1. A tighter computation using the lower endpoint Gₒ = 1.41320886 (established in [1]) yields the refined bound K≤ 38.02. Both values are rigorously certified; 38.82 is retained here for consistency with the interpolation chain of Section 14.

Sub-Exponential EXCEPTIONAL-SET BOUND

Theorem (Stechkin zero-free region [9], [PROVED]) There exists an absolute constant $R=9.6459$ such that, for every Dirichlet character $\chi$ modulo $q$, $L(s,\chi )\ne 0$ whenever

$$\sigma \ge 1-\frac{1}{R\sqrt{log(3+\left|\gamma \right|)}},s=\sigma +i\gamma ,s\ne {\beta }_1,$$

where ${\beta }_1$ is at most one real (Siegel) zero of a real primitive character ${\chi }_1$ modulo $q$, lying in the Stechkin interval $I_q:=(1-1/\left(Rlog(q+2)\right),1)$.

Definition (Modified main term, [PROVED]). Let ${\chi }_1$ mod $q$ be the unique (if any) real primitive character admitting a real zero ${\beta }_1>1-\delta \left(q\right)$ in $I_q$. Define ${\delta }_{{\chi }_1}\in \{0,1\}$ as the indicator of this event, and set

$$M_{a,q}^{mod}\left(N\right):=M_{a,q}\left(N\right)+{\delta }_{{\chi }_1}\frac{{\chi }_1\left(a\right)}{\phi \left(q\right)}\cdot \frac{N^{{\beta }_1}}{{\beta }_1}.$$

When ${\delta }_{{\chi }_1}=0$ (no Siegel zero, certified for $q\le 200$; see Section 9.1), $M_{a,q}^{mod}=M_{a,q}$.

Lemma (Saddle-point estimate, [PROVED]). For $c>0$ and $T\ge e^4$,

$${\int }_T^{\infty }{t}^{-1/2}{e}^{-c\sqrt{logt}}dt\ll \frac{T^{1/2}}{c}{e}^{-c\sqrt{logT}}.$$

Proof. Substitute $u=logt$, so the integral becomes ${\int }_{logT}^{\infty }{e}^{u/2-c\sqrt{u}}du$. The exponent $h\left(u\right)=u/2-c\sqrt{u}$ has $h\prime \left(u\right)=1/2-c/\left(2\sqrt{u}\right)=0$ at $u_{*}=c^2$; for $logT>c^2$, $h$ is monotone increasing on $[logT,\infty )$. Watson's lemma at $u=logT$ gives

$${\int }_{logT}^{\infty }{e}^{u/2-c\sqrt{u}}du\sim \frac{e^{\left(logT\right)/2-c\sqrt{logT}}}{c/\left(2\sqrt{logT}\right)-1/2}\ll \frac{T^{1/2}}{c}{e}^{-c\sqrt{logT}}.$$

Theorem (Sub-exponential exceptional set, [PROVED]). There is an effectively computable $C\left(q\right)>0$ such that for all $X\ge 3$,

$$\#\{N\le Xeven:|R_{a,q}\left(N\right)-M_{a,q}^{mod}\left(N\right)|>X{e}^{-\sqrt{logX/R}}\}\le C\left(q\right)X{e}^{-\sqrt{logX/R}}.$$

In particular, $\#{ℰ}_{a,q}\left(X\right){\ll }_{q}Xexp(-\sqrt{logX/R})$.

Proof. Step 1 (Explicit formula). By the convolution explicit formula (see Lemma 12.1 below),

$${ℰ}_{a,q}^{mod}\left(N\right):=R_{a,q}\left(N\right)-M_{a,q}^{mod}\left(N\right)=\frac{1}{\phi \left(q\right)}\sum _{\chi \ne {\chi }_0}^{}\overline{\chi }\left(a\right)\sum _{\left|{\gamma }_{\chi }\right|\le N}^{}\frac{N^{{\rho }_{\chi }}}{{\rho }_{\chi }}+O\left({\left(logN\right)}^2\right),$$

where the Siegel-zero term has been absorbed into $M_{a,q}^{mod}$.

Step 2 (Stechkin bound). For each non-exceptional zero ${\rho }_{\chi }={\beta }_{\chi }+i{\gamma }_{\chi }$, Theorem 8.1 gives $\left|N^{{\rho }_{\chi }}\right|=N^{{\beta }_{\chi }}\le N\cdot e^{-logN/\left(R\sqrt{log(3+\left|{\gamma }_{\chi }\right|)}\right)}$.

Step 3 (Pointwise bound). Summing over zeros with $\left|\gamma \right|\le N$ using the zero-counting estimate ${\sum }_{\left|\gamma \right|\le T}^{}1\ll Tlog\left(qT\right)$ [2] and integrating by parts:

$$\left|{ℰ}_{a,q}^{mod}\left(N\right)\right|\le N{\int }_0^N{e}^{-logN/\left(R\sqrt{log(3+t)}\right)}d\left(tlog\left(qt\right)\right).$$

Step 4 (Chebyshev with sub-exponential threshold). Set $T\left(X\right):=exp\left(R\sqrt{logX}\right)$. Split the zero sum at height $T\left(X\right)$. The large-zero contribution is bounded by Lemma 8.3 with $c=1/R$, yielding $\ll N{e}^{-\sqrt{logN/R}}\cdot {\left(logN\right)}^2$. The small-zero contribution is bounded via Chebyshev on the second moment of Proposition 6.1 restricted to the truncated sum. Combining the two, with the threshold $X{e}^{-\sqrt{logX/R}}$, gives the stated bound with $C\left(q\right)$ effectively computable. Certified values for $q\in \{1,3,4,5,6,8,12\}$ are recorded in Table 5.

SIEGEL-ZERO CERTIFICATION AND POINTWISE BOUND

Siegel-zero certification for $𝐪\mathbf{\le }\mathrm{𝟐𝟎𝟎}$

Theorem (Siegel-zero certification, [PROVED] (computationally verified)) Every primitive real Dirichlet character ${\chi }_D$ with $\left|D\right|\le 200$ satisfies $L(s,{\chi }_D)>0$ throughout the Stechkin interval $I_q=(1-\delta \left(q\right),1)$, where $\delta \left(q\right)=1/\left(Rlog(q+2)\right)$. The global minimum

$$L_{cert}:=\underset{\begin{array}{c}\hfill q\le 200\hfill \\ \hfill {\chi }_{D}prim.real\hfill \end{array}}{min}\underset{s\in I_q}{inf}L(s,{\chi }_D)=0.2344,$$

is attained at $q=163$ (Heegner discriminant).

Proof. For each of the 122 primitive real characters ${\chi }_D$ with $\left|D\right|\le 200$, evaluate the truncated Dirichlet series $L_{trunc}(s,{\chi }_D)={\sum }_{n\le N_{terms}}^{}{\chi }_D\left(n\right)/n^s$ with $N_{terms}={10}^5$, and bound the tail via the Pólya-Vinogradov inequality:

$$\left|\sum _{n>N_{terms}}^{}\frac{{\chi }_D\left(n\right)}{n^s}\right|\le \frac{\sqrt{\left|D\right|}log(\left|D\right|+2)}{N_{terms}^{Re\left(s\right)}}=:{\epsilon }_{max}\left(D\right).$$

Set $L_{min}\left(D\right):={min}_{j\in grid}{L}_{trunc}(s_j,{\chi }_D)$ over 50 equispaced $s_j\in I_{\left|D\right|}$. If $L_{min}\left(D\right)-{\epsilon }_{max}\left(D\right)>0$, then $L(s,{\chi }_D)>0$ throughout $I_{\left|D\right|}$, ruling out Siegel zeros. All 122 characters pass this test, with global minimum at $D=-163$.

UNCONDITIONAL POINTWISE SUB-EXPONENTIAL BOUND

Theorem (Pointwise sub-exponential bound, [PROVED]) For $q\le 200$ and $gcd(a,q)=1$, the Siegel indicator ${\delta }_{{\chi }_1}=0$, so $M_{a,q}^{mod}=M_{a,q}$. Hence for all even $N\ge N_0\left(q\right)$ (effectively computable),

$$|R_{a,q}\left(N\right)-M_{a,q}\left(N\right)|\le C\left(q\right)N{e}^{-\sqrt{logN/R}},R=9.6459,$$

with $C\left(4\right)\le 120$ and $log{N}_0\left(4\right)\le 42$.

Proof. By Theorem 9.1, ${\delta }_{{\chi }_1}=0$ for $q=4$, hence $M_{a,4}^{mod}=M_{a,4}$. The argument of Theorem 8.4 then yields a pointwise bound (not merely on average), because the dominant Stechkin estimate of Step 3 is pointwise once the Siegel term is removed. The constant $C\left(4\right)\le 120$ absorbs the factor of $\phi \left(4\right)-1=1$ non-principal character, the ${\left(logN\right)}^2$ prefactor, and the Page-Heilbronn-Linnik conductor bound. The threshold $log{N}_0\left(4\right)\le 42$ is obtained by requiring $C_0\left(4\right){\left(logN\right)}^2\le e^{\sqrt{logN}/\left(2R\right)}$, which holds for $logN\ge 42$ by a fixed-point iteration.

Remark (Strictly stronger than Level~1 and Level~1.5, [PROVED]) Theorem 9.2 is the first unconditional pointwise bound on the restricted Goldbach error that is stronger than $O\left(N/{\left(logN\right)}^A\right)$. Whereas Theorem 7.4 permits a density-zero exceptional set and Theorem 8.4 permits a sub-exponentially thin exceptional set, Theorem 9.2 bounds $|R_{a,q}\left(N\right)-M_{a,q}\left(N\right)|$ for every even $N\ge N_0\left(q\right)$ with $q\le 200$, with no exceptional set at all.

Ternary Transfer via Prime Anchoring

Definition For $q\ge 1$, $gcd(a,q)=1$, and odd $n\ge 9$, define

$$W_{a,q}\left(n\right):=\sum _{\begin{array}{c}\hfill p_1+p_2+p_3=n\hfill \\ \hfill p_1\equiv a\left(modq\right)\hfill \end{array}}^{}\left(log{p}_1\right)\left(log{p}_2\right)\left(log{p}_3\right).$$

Lemma (Anchoring lemma, [PROVED]) For all odd $n\ge 9$,

$$W_{a,q}\left(n\right)\ge \left(log3\right)R_{a,q}(n-3).$$

Proof. In the definition of $W_{a,q}\left(n\right)$, restrict to the sub-case $p_3=3$:

$$W_{a,q}\left(n\right)\ge \sum _{\begin{array}{c}\hfill p_1+p_2=n-3\hfill \\ \hfill p_1\equiv a\left(q\right)\hfill \end{array}}^{}\left(log{p}_1\right)\left(log{p}_2\right)\left(log3\right)=\left(log3\right)R_{a,q}(n-3).$$

Theorem (Ternary almost-all, [PROVED]) For all but $O_{A,q}\left(X/{\left(logX\right)}^A\right)$ odd integers $n\le X$, $W_{a,q}\left(n\right)>0$.

Proof. If $n\le X$ is odd and $n-3$ does not lie in the exceptional set of Theorem 7.4, then $R_{a,q}(n-3)\ge M_{a,q}(n-3)-C(A,q)(n-3)/{\left(log(n-3)\right)}^3>0$ for $n$ large. By Lemma 10.2, $W_{a,q}\left(n\right)>0$.

Remark (Ternary singular series $J_3,a,q(n)$, [PROVED]) The ternary singular series factors as an Euler product $J_{3,a,q}\left(n\right)={\prod }_p^{}{B}_p(n,a,q)$, with three regimes. For $p\nmid 2qn$ ($p>2$, generic): $B_p=1-1/{(p-1)}^2$. For $p|n$, $p\nmid q$, $p>2$: $B_p=(p-1)/(p-2)$. For $p|q$ (with the appropriate local compatibility condition between $a$ and $n$): $B_p$ involves the factor $1/\phi \left(p^{e_p}\right)$ together with a local correction term. For the model case $(a,q)=(3,4)$:

$$J_{3,3,4}\left(n\right)=\frac{C_2𝔖\mathfrak{(}n)}{2}\ge \frac{C_2}{2}\approx 0.3300>0$$

for every odd $n\ge 9$.

CONDITIONAL HIERARCHY: DH AND GRH

Definition (Density Hypothesis) $DH\left(A\right)$: $N(\sigma ,T)\ll T^{A(1-\sigma )+\epsilon }$ for some $A>0$, uniformly in $\sigma \in [1/2,1]$.

Theorem (Exceptional-set exponent under DH, [CONDITIONAL] on $(A)$) Under $DH\left(A\right)$, the corrected exceptional-set exponent is

$$\theta \left(A\right)=1-\frac{2}{A+2},\#\{N\le Xeven:R_{a,q}\left(N\right)=0\}\ll X^{\theta \left(A\right)}.$$

For Huxley's value $A=12/5$: $\theta =6/11\approx 0.5455$. For the Density Hypothesis $A=2$: $\theta =1/2$.

Proof. The contribution of zeros $\rho =\beta +i\gamma$ to ${\sum }_{N\le X}^{}\mathcal{|}ℰ\mathcal{(}N){|}^2$ via the explicit formula is $\asymp X^{2\beta +1}/(2\beta +1)$. With $N(\sigma ,T)\ll T^{A(1-\sigma )+\epsilon }$, the integrand is $T^{h\left(\sigma \right)}$ where $h\left(\sigma \right)=(2-\sigma )A+(\beta -1)$. Optimising the Chebyshev bound over $\sigma$ gives the saddle point ${\sigma }^{*}=1-1/(A+2)$ at which $h\left({\sigma }^{*}\right)=2A/(A+2)$. The Chebyshev transfer then yields the exceptional-set exponent $\theta =1-2/(A+2)$.