Let denote the number of positive prime numbers that are less than or equal to . The prime number theorem, one of the most celebrated results in analytic number theory, states that

.

This is a very difficult result to prove. However, Gamma (page 172) provides a heuristic argument that suggests that this answer might be halfway reasonable.

Consider all of the integers between and .

About half of these numbers won’t be divisible by 2.

Of those that aren’t divisible by 2, about two-thirds won’t be divisible by 3. (This isn’t exactly correct, but it’s good enough for heuristics.)

Of those that aren’t divisible by 2 and 3, about four-fifths won’t be divisible by 5.

And so on.

If we repeat for all primes less than or equal to , we can conclude that the number of prime numbers less than or equal to is approximately

Though not a formal proof, it’s a fast way to convince students that the unusual fraction ought to appear someplace in the prime number theorem.

When I researching for my series of posts on conditional convergence, especially examples related to the constant , 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.

I'm a Professor of Mathematics and a University Distinguished Teaching Professor at the University of North Texas. For eight years, I was co-director of Teach North Texas, UNT's program for preparing secondary teachers of mathematics and science.
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