Fix some number N. What fraction of the integers less than or equal to N are prime?

Thinking about it, we know that primes occur less and less often as N grows. Can we quantify this somehow?

Let Pi(N) denote the number of primes less than or equal to N that are prime. Then we expect that the fraction Pi(N)/N must change (decrease?) with N. In fact there is an amazing theorem called the Prime Number Theorem which says that

Pi(N)/N is asymptotic to 1/ln(N)

which means that the ratio of those two quantities approaches 1 as N goes to infinity! Thus Pi(N) is closely approximated by N/ln(N). In fact, a better estimate for Pi(N) is that it is very closely approximated by this integral:

INTEGRAL_{2 to x} dt/ln(t) .

**Presentation Suggestions:**

Write out a table of Pi(N) for the first few values of N (or flash a transparency) just to give students a concrete feel for this function before telling them the answer.

**The Math Behind the Fact:**

The proof of the Prime Number Theorem requires some hard asymptotic analysis. Several people have proved various versions of the Prime Number Theorem; among them Chebyshev, Hadamard, de la Vallee Poussin, Atle, Selberg, although the theorem was suspected by Gauss (1791).

**How to Cite this Page:**

Su, Francis E., et al. “Prime Number Theorem.” *Math Fun Facts*. <https://www.math.hmc.edu/funfacts>.

**References:**

Any book on analytic number theory

**Fun Fact suggested by: **

Francis Su