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 $$R_{a,q}(N) := \sum_{\substack{p_1 + p_2 = N \\ p_1 \equiv a \, (\mathrm{mod}\, q)}} (\log p_1)(\log p_2), \quad q \ge 1,\ \gcd(a,q)=1,$$ with expected main term $$M_{a,q}(N) := C_2\, \mathfrak{S}(N)\, \frac{N}{\varphi(q)},$$ and exceptional set $$\mathcal{E}_{a,q}(X) := \{N \le X,\ N \text{ even} : R_{a,q}(N)=0\}.$$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 $$K = 2C(1/4) \le 38.82.$$Level 1.5. A sub-exponential exceptional-set bound $$\#\mathcal{E}_{a,q}(X) \ll_{\epsilon} X \exp\!\left(-\sqrt{\frac{\log X}{R}}\right),$$ 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 $$K_{\mathrm{new}} \le 9.80,$$ and, for moduli $q \le 200$ certified free of Siegel zeros, an unconditional pointwise sub-exponential bound with $$C(4) \le 120, \quad \log N_0(4) \le 42.$$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 $\theta(A)=1 - \frac{2}{A+2}$ 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 $$N_0(4) \le 10^{16}.$$