Euler perform within the context of q-series and partitions

This morning I wrote a publish in regards to the likelihood {that a} random matrix over a finite subject is invertible. If the sector has q components and the matrix has dimensions n × n then the likelihood is

In that publish I made remark that p(q, n) converges in a short time as a perform of n [1]. One technique to see that the convergence is fast is to notice that

and

John Baez identified within the feedback that p(q, ∞) = φ(1/q) the place φ is the Euler perform.

Euler was extraordinarily prolific, and lots of issues are named after him. A number of capabilities are often known as Euler’s perform, the most typical being his totient perform in quantity concept. The Euler perform we’re interested by right here is

for −1 < x < 1. Often the argument of φ is denoted “q” however that might be complicated in our context as a result of our q, the variety of components in a subject, is the reciprocal of Euler’s q, i.e. x = 1/q.

Euler’s id [2] (on this context, to not be confused with different Euler identities!) says

This perform is straightforward to calculate as a result of the collection converges in a short time. From the alternating collection theorem now we have

When q = 2 and so x = 1/2, N = 6 is sufficient to compute φ(x) with an error lower than 2⁻⁷⁰, past the precision of a floating level quantity. When q is bigger, even fewer phrases are wanted.

For example this, now we have the next Python script.

def phi(x, N):
    s = 0
    for n in vary(-N, N+1):
        s += (-1)**n * x**((3*n**2 - n)/2)
    return s

print(phi(0.5, 6))

Each digit within the output is right.

Associated posts

[1] I didn’t say that explicitly, however I identified that p(2, 8) was near p(2, ∞).

[2] This id is also called the pentagonal quantity theorem due to its connection to pentagonal numbers.