# Prime Number Theorem

• Feb 10th 2007, 05:59 AM
AfterShock
Prime Number Theorem
Definition: Pi(x) is number of primes less than or equal to x.

Prove:

lim(x->infinity) Pi(x)/[x/ln(x)] = 1

Then, obviously for large x, Pi(x) ~ x/ln(x)

This one stumped me.
• Feb 10th 2007, 08:25 AM
CaptainBlack
Quote:

Originally Posted by AfterShock
Definition: Pi(x) is number of primes less than or equal to x.

Prove:

lim(x->infinity) Pi(x)/[x/ln(x)] = 1

Then, obviously for large x, Pi(x) ~ x/ln(x)

This one stumped me.

I believe it stumped quite a few people:D . See Hardy&Wright 5th ed pp359-367.

RonL
• Feb 10th 2007, 02:22 PM
ThePerfectHacker
Quote:

Originally Posted by AfterShock
Definition: Pi(x) is number of primes less than or equal to x.

Prove:

lim(x->infinity) Pi(x)/[x/ln(x)] = 1

Then, obviously for large x, Pi(x) ~ x/ln(x)

This one stumped me.

My mathematics advisor wrote a popular and succesful book on Complex Analysis. In the end of the book he shows where complex variables can be applied, one of problems solved is the prime number theorem. But I do not think you will understand it, it is a graduate textbook. Thus, I will agree with CaptainBlank that Hardy and Wright offer a more elementary proof, I never seen it but I know they have an elementary proof there.
• Feb 10th 2007, 06:54 PM
CaptainBlack
Quote:

Originally Posted by ThePerfectHacker
My mathematics advisor wrote a popular and succesful book on Complex Analysis. In the end of the book he shows where complex variables can be applied, one of problems solved is the prime number theorem. But I do not think you will understand it, it is a graduate textbook. Thus, I will agree with CaptainBlank that Hardy and Wright offer a more elementary proof, I never seen it but I know they have an elementary proof there.

I beleive it is Selberg's "Elementary Proof" of ca 1948 which he got into a dispute with Erdős over.

RonL
• Feb 10th 2007, 07:04 PM
ThePerfectHacker
Quote:

Originally Posted by CaptainBlank
I beleive it is Selberg's "Elementary Proof" of ca 1948 which he got into a dispute with Erdős over.

RonL

I think you are wrong.
I will quote my number theory textbook.

Quote:

Originally Posted by Elementary Number Theory by David Burton
Until recent times, the opinion prevailed that the Prime Number Theorem could not be proved without the help of the properties of the zeta function, and without recourse to complex function theory. It came as a great supprise when in 1949 the Norwegian mathematician Atle Selberg discovered a purely arithmetical proof. His paper Elementary Proof of the Prime Number Theorem is "elementary" in the technical sense of avoiding the methods of modern analysis; indeed, its content is exceedingly difficult. Selberg was awarded the Fields Medal at the 1950 International Congress of Mathematicians for his work in this area.

---

Why did he fight with Erdos?