Manuscript OpenAI@2026-10-07 · OpenAI.tex · remark · 6a931f8f
Remark: 023-01. An unconditional first moment for cubic Gauss sums
Manuscript map · lines 2925–2934
OpenAI@2026-10-07#rem:oai-p-023-01
Authors: OpenAI. Family 023.
Title: An unconditional first moment for cubic Gauss sums.
Abstract: 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.
This abstract is a catalogue record. It is not a verification and not a premise. It does not close O.1. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted.
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