Equidistribution of rational functions of primes mod \(q\) (Q2764524)

Let \(r(X)={f(X)\over g(X)}\), where \(f(X),g(X) \in \mathbb Z[X]\) are relatively prime, and denote by \({\mathcal P}_{r,q}\) the set of primes \(p\leq q\) such that \(g(p)\not\equiv 0\pmod q\). In this paper the authors prove that when \(r(X)\) is not a linear polynomial, the finite sequence \(\{ {r(p)\pmod q\over q}: p\in {\mathcal P}_{r,q}\}\) becomes uniformly distributed modulo \(1\) as \(q\to\infty\). The example \(r(X) ={3X+1\over 2}\) is noted to show that the result need not hold when \(r(X)\) is linear. For the proof the discrepancy from equidistribution is bounded in terms of an exponential sum by the Erdős-Turán inequality. Then a sieve result of Duke, Friedlander and Iwaniec is used to bound the relevant exponential sum. The main work in the paper is to show that the conditions for the applicability of the sieve result are met. The result is implicit in a paper of \textit{E. Fouvry} and \textit{P. Michel} [Ann. Sci. Éc. Norm. Supér. (4) 31, 93-130 (1998; Zbl 0915.11045)], in fact in a stronger form, but the paper under review achieves it in a simpler way.