Kummer’s Conjecture for Cubic Gauss Sums

for primes p ≡ 1 (mod 3) up to 500, and found that Sp/(2√p) lay in the intervals [−1,− 12 ], (− 12 , 12 ), [ 1 2 , 1] with frequencies approximately in the ratio 1 : 2 : 3. He conjectured, somewhat hesitantly, that this might be true asymptotically. Kummer’s conjecture was disproved by Heath-Brown and Patterson [4]. In order to state their result we must introduce a little notation. Let ω = exp(2πi/3) and let (∗/∗)3 be the cubic residue symbol for ZZ[ω]. For each c ∈ ZZ[ω] such that c ≡ 1 (mod 3) the cubic Gauss sum is