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:

    “The techniques of opening conversation are universal. I knew long ago and rediscovered that the best way to attract attention, help, and conversation is to be lost. A man who seeing his mother starving to death on a path kicks her in the stomach to clear the way, will cheerfully devote several hours of his time giving wrong directions to a total stranger who claims to be lost.”
    —John Steinbeck (1902–1968)

    “Now for civil service reform. Legislation must be prepared and executive rules and maxims. We must limit and narrow the area of patronage. We must diminish the evils of office-seeking. We must stop interference of federal officers with elections. We must be relieved of congressional dictation as to appointments.”
    —Rutherford Birchard Hayes (1822–1893)

    “I have always felt that the real purpose of government is to enhance the lives of people and that a leader can best do that by restraining government in most cases instead of enlarging it at every opportunity.”
    —Gerald R. Ford (b. 1913)

    “And why do you cry, my dear, why do you cry?
    It is all in the whirling circles of time.
    If millions are born millions must die,”
    —Robinson Jeffers (1887–1962)