{"id":"openai-math/An-unconditional-first-moment-for-cubic-Gauss-sums","pdf":"https://github.com/openai/math/blob/main/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026/paper.pdf","url":"https://github.com/openai/math/tree/main/preprints/An-unconditional-first-moment-for-cubic-Gauss-sums-September-25-2026","tags":["analytic-number-theory","gauss-sums"],"score":0,"title":"An unconditional first moment for cubic Gauss sums","domain":"Number theory","family":"023","origin":"ai-generated","source":"openai-math","authors":["OpenAI"],"reviews":0,"subject":"math.NT","summary":"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.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-09-25","reactions":{},"formalized":false,"family_title":"Patterson's first moment for cubic Gauss sums"}