Circles, the Basel Problem, and the Apparent Brightness of Stars

On Pi Day 2016, I wrote in this post about the remarkable fact, discovered by Euler, that the probability that two randomly chosen integers have no prime factors in common is \frac{6}{\pi^2}.  The proof makes use of the  famous identity \sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}, often referred to as the “Basel problem”, which is also due to Euler.  In the 2016 post I presented Euler’s original solution to the Basel problem using the Taylor series expansion for \frac{\sin(x)}{x}.

In honor of Pi Day 2018, I’d like to explain a simple and intuitive solution to the Basel problem due to Johan Wästlund.  (Wästlund’s paper is here; see also this YouTube video, which is where I first heard about this approach – thanks to Francis Su for sharing it on Facebook!)  Wästlund’s approach is motivated by physical considerations (the inverse-square law which governs the apparent brightness of a light source) and uses only basic Euclidean geometry and trigonometry.

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