Why 6/π² Appears When Counting Visible Lattice Points (and Why π² Is Actually From Earth)

 

A lattice point is a point in the Cartesian plane whose - and -coordinates are both integers.

Suppose you are standing at the origin on an infinite grid. You can see infinitely far in all directions, but you cannot see through other grid points: a point is hidden from view if some other lattice point lies along the line of sight. What fraction of the grid points are visible?

This is a problem concerning the distribution of visible lattice points from a fixed point, namely the origin.

If on the segment joining  and  lies another lattice point , then there exists  ,  such that  and  leading to . Hence a visible from the origin lattice point  satisfies

It turns out that the converse is also true. Thus,

A lattice point  is visible from the origin if and only if .

The Setup

We’re thinking of picking two random integers  (say large and uniformly distributed).

We ask:

What is the probability that ?

Call that probability .

We’ll show

Independence over primes

The key idea is:

 means no prime divides both  and .

Now fix a prime .

What’s the probability that  divides both  and ?

Since  is equaly likely to take any residue  and  is one out of

( means  divides ).

So (approximately independently, for large random integers)

Thus, for this single prime,

Multiply over all primes

To be coprime, no prime can divide both. For independent events across all primes:

This infinite product converges (because  is summable), and we can express it compactly.

Enter the zeta function

Euler’s product formula for the Riemann zeta function says:

Take :

 

So our probability is

Τhe  appears because of the Basel problem:

Euler’s proof of that identity is one of the earliest bridges between discrete arithmetic and continuous geometry (through trigonometric or Fourier analysis).

📘 In my book, “The Riemann Hypothesis Revealed: A Comprehensive Guide Through Complex Analysis,” I include both Euler’s original argument and a modern contemporary proof, showing how deeply this connection runs through analytic number theory.

A geometric picture on the lattice

Imagine all lattice points in the first quadrant, and for each primitive point (visible one) draw a straight ray from the origin. The visible points represent directions — rational slopes .
Now, as  get large, the density of such visible points measures how many distinct rational directions there are per unit area.

That density being is not a coincidence.

 Roughly speaking, the  comes in because counting lattice points relates to the area of circles in the plane, and  arises as the analytic version of that counting.

Mikael Passare (see here) offers a fresh geometric proof for the Basel problem, proving that the infinite sum of squared reciprocals equals . 

The method utilizes bipolar coordinates to map triangle interior angles to logarithmic side lengths. 

Passare demonstrates that this transformation is area-preserving, allowing the total area of an "amoeba" shape in the logarithmic plane to equal a specific triangle's area. By interpreting the series as stacked exponential curves that fill this region, the proof elegantly connects basic trigonometry and calculus to resolve a centuries-old mathematical puzzle.

So next time you see π² in a formula, don’t call it otherworldly - it’s just geometry doing its job on a very earthly lattice.