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.

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

Published by John Quintanilla

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Published by John Quintanilla

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