On a Conjecture of Helleseth

We are concern about a conjecture proposed in the middle of the seventies by Hellesseth in the framework of maximal sequences and theirs crosscorrelations. The conjecture claims the existence of a zero outphase Fourier coefficient. We give some divisibility properties in this direction. 1. Two conjectures of Helleseth Let L be a finite field of order q > 2 and characteristic p. Let μ be the canonical additive character of L i.e. μ(x) = exp(2iπTr (x)/p) where Tr is the trace function with respect to the finite field extension L/Fp. The Fourier coefficient of a mapping f : L → L is defined at a ∈ L by (1) f̂(a) = ∑ x∈L μ(ax+ f(x)). The distribution of these values is called the Fourier spectrum of f . Note that when f is a permutation the phase Fourier coefficient f̂(0) is equal to 0. The mapping f(x) = x is called the power function of exponent s, and it is a permutation if and only if (s, q − 1) = 1. Moreover, if s ≡ 1 mod (p − 1) the Fourier coefficients of f are rational integers. Helleseth made in [3] two “global” conjectures on the spectra of power permutations. The first claims the vanishing of the quantity (related to Dedekind determinant, see [9]) (2) D(f) = ∏