An infinite product challenge

Gil Kalai wrote a blog post yesterday entitled “Test Your Intuition (or knowledge, or programming skills) 36.” The challenge is to evaluate the infinite product

$\prod_{p\,\, \mathrm{prime}} \frac{p^2+1}{p^2 - 1}$

I imagine there’s an elegant analytical solution, but since the title suggested that programming might suffice, I decided to try a little Python. I used primerange from SymPy to generate the list of primes up to 200, and cumprod from NumPy to generate the list of partial products.

    cumprod(
        [(p*p+1)/(p*p-1) for p in primerange(1,200)]
    )

Apparently the product converges to 5/2, and a plot suggests that it converges very quickly.

Plot of partial products

Here’s another plot to look more closely at the rate of convergence. Here we look at the difference between 5/2 and the partial products, on a log scale, for primes less than 2000.

Plot of 2.5 minus partial products, log scale

One thought on “An infinite product challenge”

GlennF

5 March 2019 at 10:25

(brief proof sketch) Rearrange the terms so that they are in terms of powers of 1/p. Then notice you can multiply the top and the bottom by (1-(1/p^2)). The answer becomes recognizable from the product formula for the Riemann zeta function as \zeta(2)^2 / zeta(4) = 90/36 = 5/2.

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