The prime number theorem in Calculus II

I attended Shahriar Shahriari’s MAA Minicourse Beyond Formulas and Algorithms: Teaching a Conceptual/thematics Single Variable Calculus Course at the 2008 Joint Mathematics Meeting. He talked about having his calculus students derive the prime number theorem.

Recall that the prime number theorem states that if $\pi(x)$ is the number of primes less than or equal to $x$, then

$\displaystyle\lim_{x\to\infty}\frac{\pi(x)}{x/\ln(x)}=1$.

He inspired my to try this in my Calculus II class. I gave my class the following worksheet, which they completed during a lab period. It went well, and I thought others might be interested in it.

• Here is a pdf of the prime number theorem lab. It has them discover that $\displaystyle\pi(x)\approx\frac{x}{\ln x+1.08}$.
• Here is a pdf of the solutions.

I also borrowed heavily from the material in Shahriar Shahriari’s book Approximately Calculus.

[God plays dice recently posted a heuristic derivation of the prime number theorem.]