Restricted Goldbach Sums in Arithmetic Progressions: Local Obstructions, an Explicit Almost-All Framework, and a General-Modulus Extension

Source abstract

For $q≥1$ and $gcd\left(a,q\right)=1$ consider the restricted weighted Goldbach sum $R_{a,q}(N) :=\sum_{\substack{p_1+p_2=N \\ p_1 \equiv a \ (\mathrm{mod}\ q)}} \left({\log{p}}_{1}\right)\left({\log{p}}_{2}\right)\\$, the sum over primes. We give a fully rigorous, self-contained treatment of the restricted binary, ternary, and quaternary Goldbach problem in this setting. We first prove an elementary but decisive local obstruction: if an odd prime $l$ divides $q$, then on a positive-density set of even $N$ the sum $R_{a,q}(N)$ collapses to $O\left(logN\right)$, so no main term of size $\asymp N/\varphi \left(q\right)$ can hold uniformly; for powers of two $q=2^{k}$ this obstruction is absent, and we identify the correct local main term $M_{a,q}(N)=\frac{2C_{2}}{\varphi (q)}S(N)N$. Within this scope we prove a qualitative almost-all theorem, with a complete major/minor-arc derivation, for fixed $q=2^{k}$. We then resolve, for every fixed modulus $q≥1$ (not merely powers of two), the correct corrected local main term and the associated almost-all theorem: writing $S\left(Nq\right)$ for the twin-prime-type Euler factor evaluated at the literal product $Nq$, we prove $\sum_{N\leq X,\,2\mid N} \left|R_{a,q}\left(N\right)-1_{\gcd{\left(N-a,q\right)}=1}\,\frac{N}{\varphi \left(q\right)}\,2C_{2}S\left(Nq\right)\right|{\ll }_{q}X^{2}(logX)^{-A}$ for every $A>0$, closing what an earlier stage of this programme recorded as an open problem on general moduli. We record a conditional (hypothesis-explicit) impossibility theorem showing that no threshold independent of $X$ can upgrade the fixed-modulus almost-all theorem to an effective almost-all statement once a matching second-moment lower bound is granted, and we prove positivity of the restricted quaternary singular series through explicit local densities. We isolate, as honestly labelled structural cautions rather than theorems, the four classical routes (Siegel-zero absorption, Borel–Cantelli divergence, discrepancy explosion, and circle-method saturation at GRH) that fail to upgrade the almost-all theorems to unconditional finiteness of the exceptional set. Throughout we distinguish carefully between what the classical circle method establishes and stronger statements (pointwise asymptotics, explicit numerical thresholds, sub-exponential exceptional-set bounds) that it does not, collecting the latter as precisely stated open problems rather than asserting them.