Restricted Goldbach Sums in Arithmetic Progressions: Analytic Hierarchy, Sub-Exponential Bounds, and Riemann Zero Detection

Source abstract

We develop a unified and fully audited analytic hierarchy for the restricted weighted Goldbach sum$$R_{a,q}(N) := \sum_{\substack{p_1+p_2=N \ p_1 \equiv a \pmod{q}}} (\log p_1)(\log p_2), \qquad q \geq 1,\ \gcd(a,q)=1,$$with expected main term $M_{a,q}(N) := C_2, S(N), N/\varphi(q)$, and exceptional set $E_{a,q}(X) := {N \leq X,\ N\ \text{even} : R_{a,q}(N) = 0}$. The paper consolidates and supersedes preprint version 3, integrating results from Papers 1, 9 and 14 of the Anderson Series, with all documented corrections applied.The unconditional core establishes three nested levels. Level 1 is an effective almost-all theorem via the standard $L^4$ minor-arc route, with explicit constant $K = 2C(1,4) \leq 38.82$. Level 1.5 is a sub-exponential exceptional-set bound $\#\mathcal{E}_{a,q}(X) \ll_{q} X \exp\left(-\frac{\sqrt{\log X}}{R}\right)$ with Stechkin's constant $R = 9.6459$, proved unconditionally by absorbing any potential Siegel zero into a modified main term. Level 1.5+ is a Hölder minor-arc refinement giving the improved constant $K_{\text{new}} \leq 9.80$ and, for moduli $q \leq 200$ certified free of Siegel zeros, an unconditional pointwise sub-exponential bound with $C(4) \leq 120$ and $\log N_0(4) \leq 42$.Three structural obstructions (Double-Pole, Borel–Cantelli, ETK Dimensional Explosion) formally retract three classical routes to unconditional finiteness. Under DH and GRH, conditional hierarchies (with $\theta(A) = 1 - 2/(A+2)$ and $\log N_0(4) = 45.93$) are recorded. The Gowers–Spectral Bridge gives conditional finiteness under the Uniform Spectral Gap (USG) hypothesis with effective threshold $N_0(4) \leq 10^{16}$.The open sub-lemma (Proposition 13.5) connecting USG to the Montgomery Pair Correlation Conjecture has been resolved conditionally. The complete logical chain $\text{Strong Montgomery-GUE} \Rightarrow \text{USG} \Rightarrow R_{a,q}(N) > 0$ for all $N \geq N_0$ is now fully established (conditional on Strong Montgomery-GUE, which is strictly stronger than the standard weak form of Conjecture 13.10). See Sections 13.5–13.6.[HONEST CAVEAT] The proof of Theorem 13.15 invokes EAC (Controlled Additive Energy) as an intermediate step. By Theorem 13.16, EAC is equivalent to Strong Montgomery-GUE, which is strictly stronger than the standard Montgomery Pair Correlation Conjecture (Conjecture 13.10). The logical chain should therefore be read as: Strong Montgomery-GUE $\Rightarrow$ USG $(c=1)$ $\Rightarrow$ finiteness of $E_{a,q}$. The standard weak Montgomery conjecture alone does not suffice; see Section 13.6, Obstacle 1.