On the distribution of the roots of polynomial congruences

Abstract

In a recent paper on a divisor problem the author showed incidentally that there is a certain regularity in the distribution of the roots of the congruencefor variable k, where D is a fixed integer that is not a perfect square. In fact, to be more precise, it was shown that the ratios v/k, when arranged in the obvious way, are uniformly distributed in the sense of Weyl. In this paper we shall prove that a similar result is true when the special quadratic congruence above is replaced by the general polynomial congruencewhere f(u) is any irreducible primitive polynomial of degree greater than one. An entirely different procedure is adopted, since the method used in the former paper is only applicable to quadratic congruences.