On the location of the roots of polynomial congruences

Abstract

We have indicated in our tract [9] that several interesting problems in the theory of numbers are related to results about the evenness of the distribution of the roots v of a polynomial congruencewhere f(x) = a0xn + … + an is an irreducible polynomial having integral coefficients and degree n≧2. We alluded, for example, to our work on the Chebyshev problem of the greatest prime factor of n2 – D [8], in which an essential component was our earlier demonstration [6] of the uniform distribution, modulo 1, of v/k when f(x) = x2 – D. But, having pointed out that the quantitative descriptions of such uniformity had to be very sharp for substantial applications, we then noted with regret that little more than mere uniform distribution was obtained in our generalization [7] of [6] to congruences of higher degree. Indeed, it has only been for certain cubic polynomials that results have been produced that are comparable in power with those for quadratic polynomials, and even these depend on the assumption of the unproved hypothesis R* regarding the size of incomplete Kloosterman sums [10].