Pair correlation of Farey fractions with square-free denominators (Q6619738)

The study of Farey fractions \(F_Q = \{a/q: q \leq Q, \gcd(a,q) = 1\}\) has a rich history in number theory that is partially due to the Franel-Landau equivalence that connects the (fine-scale) distribution of the Farey sequence with the Riemann hypothesis.\N\NOne way to study fine-scale distribution in general, is looking at the correlations of sequences. This has led recently to a large body of research articles, in particular in the most attainable case of pair correlations: Here, given a sequence \((x_n)_{n \in \mathbb{N}}\), we say that a sequence \((x_n)_{n \in \mathbb{N}} \subseteq [0,1)\) has a limiting pair correlation function \(g: [0,\infty) \to [0,\infty)\) if for any interval \(I \subseteq [0,\infty)\), we have\N\[\N\lim_{N \to \infty} \frac{1}{2N}\#\left\{1 \leq n \neq m \leq N: \lVert x_n- x_m \rVert \in \frac{I}{N} \right\} = \int_{I} g(x) \,\mathrm{d}x.\N\]\NWhile the Poissonian case (that is, \(g \equiv 1\)) that holds almost surely for a uniformly distributed i.i.d. sample and has been proven for several number-theoretic sequences, the behaviour for Farey sequences deviates from the Poissonian case: \textit{F. P. Boca} and \textit{A. Zaharescu} [J. Lond. Math. Soc., II. Ser. 72, No. 1, 25--39 (2005; Zbl 1089.11037)] showed that in this case, the limiting pair correlation function exists and is defined by \N\[\Ng(x) = \frac{6}{\pi^2x^2}\sum_{1 \leq k < \pi^2x/3}\varphi(k)\log \frac{\pi^2x}{3k},\tag{1}\N\]\Nwhere \(\varphi\) denotes the Euler totient function. While this shows a strong repulsion phenomenon when \(x\) is small, the picture changes when considering only prime denominators: In this case, \textit{M. Xiong} and \textit{A. Zaharescu} [J. Number Theory 128, No. 10, 2795--2807 (2008; Zbl 1219.11114)] proved the Poissonian behaviour.\N\NThus the question arises at which point the phenomenon changes when being between the prime denominators (Poissonian) and taking all denominators (strong repulsion). In this article, the corresponding pair correlation function is studied for the case when restricting to square-free denominators: The main result (Theorem 1.1) proves that a limiting pair correlation function \(g\) exists and the explicit function is given in a form similar to (1), with the main difference being that \(\varphi\) is replaced by another (quite complicated) multiplicative function. In particular, again \(g(x) = 0\) for small \(x\) and thus, although the limiting pair correlation function deviates from the Farey case, the strong repulsion phenomenon still holds for the square-free case.\N\NThe proof follows the classical paths, with a main ingredient being exponential sums.