On the distribution of polynomial Farey points and Chebyshev's bias phenomenon

Abstract

We study two types of problems for polynomial Farey fractions. For a positive integer $Q$, and polynomial $P(x)\in\mathbb{Z}[X]$ with $P(0)=0$, we define polynomial Farey fractions as \[\mathcal{F}_{Q,P}:=\left\{\frac{a}{q}: 1\leq a\leq q\leq Q,\ \gcd (P(a),q)=1\right\}.\] The classical Farey fractions are obtained by considering $P(x)=x$. In this article, we determine the global and local distribution of the sequence of polynomial Farey fractions via discrepancy and pair correlation measure, respectively. In particular, we establish that the sequence of polynomial Farey fractions is uniformly distributed modulo one and show that the limit superior of the pair correlation measure of $(\mathcal{F}_{Q,P})_{Q\ge1}$ is bounded. For the specific polynomial $P(x)=x(x+1)$, we show the existence of the limiting pair correlation measure of $(\mathcal{F}_{Q,P})_{Q\ge1}$ and also provide an explicit formula for the pair correlation function which is non-Poissonian. Further, restricting the polynomial Farey denominators to certain subsets of primes, we explicitly find the pair correlation measure and show it to be Poissonian. Finally, we study Chebyshev's bias type of questions for the classical and polynomial Farey denominators along arithmetic progressions and obtain an $Ω$-result for the error term of its counting function.