First Moment of Distances Between Centres of Ford Spheres

Abstract

This paper aims to develop the theory of Ford spheres in line with the current theory for Ford circles laid out in a recent paper by S. Chaubey, A. Malik and A. Zaharescu. As a first step towards this goal, we establish an asymptotic estimate for the first moment $\mathcal{M}_{1,I_2} (S) = \sum\limits_{\substack{\frac{r}{s},\frac{r'}{s'}\in \mathcal{G}_S \\ consec}} \frac{1}{2\lvert s \rvert ^2} + \frac{1}{2\lvert s' \rvert ^2}$ where the sum is taken over pairs of fractions associated with `consecutive' Ford spheres of radius less than or equal to $\frac{1}{2S^2}$.