What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 7

Suppose that two positive integers are chosen at random. What is the probability that they are relatively prime (that is, have no common factors except 1)?

The answer is exactly what you’d expect it be (Gamma, p. 68): $6/\pi^2$, or about 60.8%.

Yes, that was a joke.

Indeed, if $k$ positive integers are random, the probability that they are relatively prime is $1/\zeta(k)$, where Riemann’s zeta function arises once again.

Even more, the probability that $k$ random positive integers lack a $n$th power common divisor is $1/\zeta(nk)$.

I’ll refer the interested reader to Gamma and also to Mathworld (and references therein) for more details.

When I researching for my series of posts on conditional convergence, especially examples related to the constant $\gamma$, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.