Download Counting Primes (3/19)

Counting Primes (3/19)
• Given a number n, how many primes are there between
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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:
 ( n)
lim n
1
n / ln(n)
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
n
1
“logarithmic integral” Li(n) =
dx
 ln( x)
2
• Check this out in Mathematica .
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., e  lim n (1  1 ) n .
n
• For Friday, please read Chapter 13.