Patterson's first moment for cubic Gauss sums

What OpenAI claims

Proves unconditionally the all-primary-prime form of Patterson’s first-moment asymptotic: normalized cubic Gauss sums over primary Eisenstein primes of norm at most X, including both conjugates, have an explicit positive main term of order X^5/6/log X. Every fixed nonzero prime-angle Fourier mode has smaller order. (Lean)

Paper

1. An unconditional first moment for cubic Gauss sums

   25 September 2026 · Preprint, not peer reviewed · PDF

   We prove Patterson's first-moment conjecture for normalized cubic Gauss sums over all primary Eisenstein primes, unconditionally. The sharp-cutoff main term is (6/5)c_*X^5/6/log X, where c_*=(2π)^2/3/(3Γ(2/3)). For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the first-moment scale.

Status

Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.