An application of a product formula for the cubic Gauss sum

A product formula of Matthews [4] for the cubic Gauss sum τ3(ω)τ3(ω) as defined in the Introduction will be applied to determine which of the three intervals (−2p,−p), (−p,p) and (p,2p) contains the cubic Gauss sum g3(p)=∑a=0p−1e2πia3/p, where p is a prime number congruent to one modulo 3.