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Pair correlation of sums of rationals with bounded height. (English) Zbl 1218.11069
For each positive integer \(Q\) denote the Farey sequence of order \(Q\) by \(\mathcal{F}_Q\). The pair correlation measure associated to \(\mathcal{F}_Q\) was proved to converge as \(Q\to\infty\) by F. P. Boca and A. Zaharescu [J. Lond. Math. Soc., II. Ser. 72, No. 1, 25–39 (2005; Zbl 1089.11037)] who showed that the limiting measure is absolutely continuous with respect to Lebesgue measure, and provided an explicit formula for the corresponding limiting pair correlation function \[ g(\lambda)=\frac{6}{\pi^2\lambda^2}\sum_{1\leq k\leq \frac{\pi^2\lambda}{3}}\varphi(k)\log\frac{\pi^2\lambda}{3k}, \] for any \(\lambda>0\), where \(\varphi\) is the standard Euler totient function.
In this article the authors show that the limiting pair correlation function of \(\mathcal{F}_Q+\mathcal{F}_Q\) modulo \(1\) exists, as \(Q\to\infty\), on any subinterval \(\mathbf{I}\subset[0,1]\), and is given by \[ g_2(\lambda)=\frac{c}{\pi^2 \lambda^2}\sum_{1\leq k\leq \frac{\pi^4 \lambda}{9}} \Psi(k) \log^3\frac{\pi^4 \lambda}{9k}, \] for any \(\lambda>0\), where \[ c=\prod_{p\;\text{prime}}\left(1-\frac{2}{p(p+1)}\right)\left(1-\frac{3}{p(p+2)}\right), \] and \(\Psi(n)\) is given defined in terms of its generating Dirichlet series \[ \sum_{n\geq 1}\frac{\Psi(n)}{n^s}=\frac{\zeta(s-1)}{\zeta^4(s)}H_p(s) \] where \(\zeta(s)\) is the Riemann zeta function and \[ H_p(s):=\prod_{p\;\text{prime}}\left(1+\frac{(p-1)(p+4)}{p(p+3)}\left(-\frac{4}{p-1}+\sum_{k\geq 1}\frac{(1-p^{(k+1)(1-s)})^4-(1-p^{k(1-s)})^4}{(1-p^{1-s})p^{k(2-s)}}\right)\right). \]

MSC:
11J71 Distribution modulo one
11B57 Farey sequences; the sequences \(1^k, 2^k, \dots\)
62H20 Measures of association (correlation, canonical correlation, etc.)
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