OAI.CubicFirstMoment.patterson_firstMoment

The theorem states that, working in the Eisenstein integers Z[ω] with ω = exp(2πi/3), the sum over all primary Eisenstein primes p of norm N(p) = |p|² at most X of the normalized cubic Gauss sum g(p) equals (6/5)·c*·X^(5/6)/log X up to an error that is o(X^(5/6)/log X) as X tends to infinity, where c* = (2π)^(2/3)/(3Γ(2/3)). Here an element p is primary when 3 divides p − 1, and a primary prime is a primary element that is prime in Z[ω]. The Gauss sum g(p) is N(p)^(-1/2) times the sum, over residue classes v modulo p (using a chosen representative of each class), of the cubic residue symbol of v at p multiplied by the additive phase exp(2πi(z + z̄)) with z = v/p. The cubic symbol is 0 when v is not a unit modulo p; otherwise it is ω^j, where j in {0,1,2} is the index with v^((N(p)−1)/3) congruent to ω^j modulo p, and it is 0 if no such j exists. The statement is the defined proposition FirstMomentStatement, asserted here as an admitted theorem.