Patterson's first moment for cubic Gauss sums

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)

Manuscript

An unconditional first moment for cubic Gauss sums

September 25, 2026 · source · cite as \cite{OAI:An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026}

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_*=(2\pi)^{2/3}/(3\Gamma(2/3)). For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the first-moment scale.

Result 023 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.