Chebyshev's Bias in Goldbach's Comet: the Residue-Class Split of Goldbach Representations and the Zeros of L(s, χ₋₃)

Abstract

Dividing the Goldbach representation count r(N) by the Hardy-Littlewood main term removes the banding of Goldbach's comet entirely; for every even N up to 10^7 and a sample to 10^8 the ratio tends to 1 with a residual independent of the small prime factors of N. One residual survives: even N congruent to 1 mod 3 have more representations than those congruent to 2 mod 3, by c(N)/sqrt(N) with c of order 4 over three decades. We identify this as Chebyshev's bias for the modulus 3 acting on both primes of a representation and confirm numerically, zero by zero, the explicit formula that governs it: the prime-square term predicts c = 4.35 with no free parameter against an observed 4.21, and the oscillation of c(N) is reproduced in frequency, amplitude and phase by the first four nontrivial zeros of L(s, chi_{-3}), at heights 8.0397, 11.2492, 15.7046 and 18.2620; a blind frequency scan returns 8.04. The zero-parameter prediction correlates at 0.95 with forty bins of data. That such averages are governed by zeros of L-functions is a theorem (Fujii 1991; Bhowmik, Halupczok, Matsumoto and Suzuki 2019); the contribution is the comparison at the level of individual zeros with nothing fitted. We also report twin-irreducible counts over F_q[T] for q = 2, 3, 5, 7 outside the range of the Sawin-Shusterman theorem, and the first computation of Stevenhagen's generalised Redei symbol on twin prime pairs, which is blind to twinness. Nothing here bears on the truth of Goldbach's conjecture. Computations and drafting were carried out with Claude (Anthropic). Not peer reviewed.