Patterson's conjecture on the first moment of cubic Gauss sums over primes

Theorem 1.1: unconditionally, $\sum_{N\pi\le X,\ \pi\equiv1(3)\text{ prime}}G(\pi)=\frac65c_*X^{5/6}/\log X+o(X^{5/6}/\log X)$, over all primary Eisenstein primes; equivalently $\sum_{p\le X}S_p/(2\sqrt p)\sim\frac35c_*X^{5/6}/\log X$, matching Wan's Conjecture 4.13. Also cancellation in every fixed nonzero angular mode and for fixed powers $G(\pi)^k$ with $3\nmid k$, $|k|>1$. The error term is only little-oh, with no rate; powers $k$ divisible by 3 (part of Patterson's complementary conjecture) remain open. Uses the unconditional Dunn-Radziwill Voronoi formula, not their GRH estimates.

Theorem 1.1: unconditionally, $\sum_{N\pi\le X,\ \pi\equiv1(3)\text{ prime}}G(\pi)=\frac65c_*X^{5/6}/\log X+o(X^{5/6}/\log X)$, over all primary Eisenstein primes; equivalently $\sum_{p\le X}S_p/(2\sqrt p)\sim\frac35c_*X^{5/6}/\log X$, matching Wan's Conjecture 4.13. Also cancellation in every fixed nonzero angular mode and for fixed powers $G(\pi)^k$ with $3\nmid k$, $|k|>1$. The error term is only little-oh, with no rate; powers $k$ divisible by 3 (part of Patterson's complementary conjecture) remain open. Uses the unconditional Dunn-Radziwill Voronoi formula, not their GRH estimates.