Counting Primes (3/19)• Given a number n, how many primes are there between

2 and n?• No one has discovered an exact formula (and no one will!).• So, change the question: Given a number n, about how

many primes are there between 2 and n?• Let’s experiment a bit with Mathematica. We denote the

exact number of primes below n by (n).• The Prime Number Theorem (PNT). The number of primes

below n is approximated by n / ln(n). More specifically:

1)ln(/

)(lim nn

nn

Comments on the PNT• It was a huge accomplishment of 19th Century mathematics.• Another (illuminating) way to say what the PNT says is that

in the neighborhood of a number n, about 1 out of every ln(n) numbers will be primes. Or, put another way, the density of primes near n is 1 / ln(n).

• This leads us to an even better estimator for (n): the “logarithmic integral” Li(n) =

• Check this out in Mathematica .

n

dxx2 )ln(

1

More Comments• It’s absolutely astounding (to me at least) that the number

of primes below n is somehow related to the number e 2.71828.

• What is this number e anyway? Where does it come from? Where does it arise in nature?

• Well, it’s most easily described as the natural limit of compounding, i.e., .

• For Friday, please read Chapter 13.

nn ne )11(lim