Mini-Challenge #3: Answer

The value of X(100) is 3.1388241 ...

As k -> infinity, the quantity X(k) tends towards π = 3.1415926 ...   but rather slowly.

Consider the fraction j / n.  This fraction is not included in the Farey series if j and n share a common factor. It is only included if j and n are coprime. So the number of Farey fractions with denominator n is equal to the number of integers less than n which are coprime to n.

The mathematician Leonhard Euler defined a function with this property: Euler's totient function φ(n) is defined to be the number of positive integers less than n which are coprime to n. For prime numbers p, φ(p) = p - 1. This is clearly an upper bound.

The number of fractions in a Farey series of order n is:

   φ(2) + φ(3) + φ(4) + ... + φ(n).

There is a remarkable formula involving the φ function and the constant π (pi), the ratio of the circumference to the diameter of a circle. The sum

  φ(1) + φ(2) + φ(3) + φ(4) + ... + φ(n)

tends towards 3 n2 / π2 as n increases!