OpenAI research collection · 1 manuscript · Source checked 2026-10-19
OpenAI's result summary
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.
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MANUSCRIPT 01
An unconditional first moment for cubic Gauss sums
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}/(3\Gamma(2/3)). For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the…
Full source 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_*=(2π)^{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.
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