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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 . Every fixed nonzero prime-angle Fourier mode has smaller order.
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An unconditional first moment for cubic Gauss sums
Sep 25, 2026·paper.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 , where . For every fixed nonzero angular Fourier mode of the prime argument, we also prove cancellation at the first-moment scale.