Euler function

This morning I wrote a post about the probability that a random matrix over a finite field is invertible. If the field has q elements and the matrix has dimensions n × n then the probability is In that post I made observation that p(q, n) converges very quickly as a function of n [1]. One way to see that the convergence is quick is to note that and John Baez pointed out in the comments that p(q, ∞) = φ(1/q) where φ is the Euler function. Euler was extremely prolific, and many things are named after him. Several functions are known as Euler’s function, the most common being his totient function in number theory. The Euler function we’re interested in here is for −1 < x < 1. Usually the argument of φ is denoted “q” but that would be confusing in our context because our q, the number of elements in a field, is the reciprocal of Euler’s q, i.e. x = 1/q. Euler’s identity [2] (in this context, not to be confused with other Euler identities!) says This function is easy to calculate because…