What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 16

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

\pi(x) \approx \displaystyle \frac{x}{\ln x}.

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 1 and x.

  • 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 \sqrt{x}, we can conclude that the number of prime numbers less than or equal to x is approximately

\pi(x) \approx \displaystyle x \prod_{p \le \sqrt{x}} \left(1 - \frac{1}{p} \right).

From this point, we can use Mertens product formula

\displaystyle \lim_{n \to \infty} \frac{1}{\ln n} \prod_{p \le n} \left(1 - \frac{1}{p} \right)^{-1} = e^\gamma

to conclude that

\displaystyle \frac{1}{\ln n} \prod_{p \le n} \left(1 - \frac{1}{p} \right) \approx \displaystyle \frac{e^{-\gamma}}{\ln n}

if n is large. Therefore,

\pi(x) \approx x \displaystyle \frac{e^{-\gamma}}{\ln \sqrt{x}} = 2 e^{-\gamma} \displaystyle \frac{x}{\ln x}.

Though not a formal proof, it’s a fast way to convince students that the unusual fraction \displaystyle \frac{x}{\ln x} ought to appear someplace in the prime number theorem.

green line

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.

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