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:

    If education is always to be conceived along the same antiquated lines of a mere transmission of knowledge, there is little to be hoped from it in the bettering of man’s future. For what is the use of transmitting knowledge if the individual’s total development lags behind?
    Maria Montessori (1870–1952)

    Many women are reluctant to allow men to enter their domain. They don’t want men to acquire skills in what has traditionally been their area of competence and one of their main sources of self-esteem. So while they complain about the male’s unwillingness to share in domestic duties, they continually push the male out when he moves too confidently into what has previously been their exclusive world.
    Bettina Arndt (20th century)

    I am the first to admit that I am no great orator or no person that got where I have gotten by any William Jennings Bryan technique.
    —Gerald R. Ford (b. 1913)

    Think of the wonderful circles in which our whole being moves and from which we cannot escape no matter how we try. The circler circles in these circles....
    —E.T.A.W. (Ernst Theodor Amadeus Wilhelm)