On the discrepancy of uniformly distributed roots of quadratic congruences

Abstract

In this paper we consider the distribution of fractional parts { ν / p } , where p is a prime less than or equal to x and ν is the root in Z / p Z of a quadratic polynomial with negative discriminant. This set is known to be uniformly distributed as x → ∞ . Here we apply the Erdős–Turán inequality to obtain an estimate for the discrepancy.