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

Read more about this topic:  Ford Circle

Famous quotes containing the words total, area, ford and/or circles:

    “only total expression

    expresses hiding: I’ll have to say everything
    to take on the roundness and withdrawal of the deep dark:
    less than total is a bucketful of radiant toys.”
    —Archie Randolph Ammons (b. 1926)

    “Prestige is the shadow of money and power. Where these are, there it is. Like the national market for soap or automobiles and the enlarged arena of federal power, the national cash-in area for prestige has grown, slowly being consolidated into a truly national system.”
    —C. Wright Mills (1916–1962)

    “Government ... thought [it] could transform the country through massive national programs, but often the programs did not work. Too often they only made things worse. In our rush to accomplish great deeds quickly, we trampled on sound principles of restraint and endangered the rights of individuals.”
    —Gerald R. Ford (b. 1913)

    “The whole force of the respectable circles to which I belonged, that respectable circle which knew as I did not the value of security won, the slender chance of replacing it if lost or abandoned, was against me ...”
    —Ida M. Tarbell (1857–1944)