Ford Circle - Total Area of Ford Circles

Total Area of Ford Circles

There is a link between the area of Ford circles, Euler's totient function φ, the Riemann zeta function ζ, and Apéry's constant ζ(3).

As no two Ford circles intersect, it follows immediately that the total area of the Ford circles

is less than 1. In fact the total area of these Ford circles is given by a convergent sum, which can be evaluated.

From the definition, the area is

A = \sum_{q\ge 1} \sum_{ (p, q)=1 \atop 1 \le p < q } \pi \left( \frac{1}{2 q^2} \right)^2.

Simplifying this expression gives

A = \frac{\pi}{4} \sum_{q\ge 1} \frac{1}{q^4} \sum_{ (p, q)=1 \atop 1 \le p < q } 1 = \frac{\pi}{4} \sum_{q\ge 1} \frac{\varphi(q)}{q^4} = \frac{\pi}{4} \frac{\zeta(3)}{\zeta(4)},

where the last equality reflects the Dirichlet generating function for Euler's totient function φ(q). Since ζ(4) = π 4/90, this finally becomes

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