there's an identity that still amazes me every time i see it.
P(gcd(a, b) = 1) = 6/π² (approx. 0.607927)
where this probability is defined via the limit of the proportion of coprime pairs in {1, ..., N}² as N → ∞.
one heuristic for why this is true is surprisingly elegant.
for any prime p,
P(p divides both a and b) = 1/p²
so
P(p does not divide gcd(a, b)) = 1 − 1/p²
which suggests
P(gcd(a, b) = 1) = ∏ₚ (1 − 1/p²)
now euler's product formula tells us:
ζ(s) = ∏ₚ 1/(1 − p⁻ˢ)
so at s = 2
∏ₚ (1 − 1/p²) = 1/ζ(2)
and since
ζ(2) = 1 + 1/2² + 1/3² + 1/4² + ... = π²/6
we get
P(gcd(a, b) = 1) = 6/π² (approx. 0.607927)
my favorite part isn't actually the probability
it's the geometric interpretation
a lattice point (a, b) is visible from the origin if and only if
gcd(a, b) = 1
because otherwise another lattice point lies on the same line segment and blocks the view.
therefore,
density of visible lattice points = 6/π²
i still find it remarkable that the same constant simultaneously connects
- prime factorization
- the riemann zeta function
- analytic number theory
- and a purely geometric notion of visibility on the integer lattice
---
edit: corrected the wording about choosing integers "uniformly at random." thanks to those who pointed it out.