On the Distribution of Powers in Finite Fields

AbstractUsing a special ordering {x0,…,xpf−1} of the elements of an arbitrary finite field and the termsemicyclic consecutive elements, defined in Winterhof (“On the Distribution of Squares in Finite Fields,” Bericht 96/20, Institute für Mathematik, Technische Universitüt Braunschweig), some distribution properties of arbitrarynth powers are deduced. So Perron’s famous theorem on the distribution of quadratic residues is generalized: Ifχdenotes a nontrivial multiplicative character of ordern∣pf−1 andaa nonzero element ofFpf, then for allnth roots of unityω≠1 the number ofx∈Fpfwithχ(x)χ(x+a)=ωis equal to (pf−1)/n.Furthermore, bounds for incomplete character sums and for the largest numberLpfof semicyclic consecutive elements with the same character values are given. For example, the classical Polya–Vinogradov bound is generalized to∣∑ν=0χ(xa+xν)∣≤pf(1−p1−f+logpf)