Results 1 to 5 of 5

Math Help - Prime Number Theorem

  1. #1
    Senior Member
    Joined
    Apr 2006
    Posts
    401

    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.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by AfterShock View Post
    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 . See Hardy&Wright 5th ed pp359-367.

    RonL
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by AfterShock View Post
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by ThePerfectHacker View Post
    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
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by CaptainBlank View Post
    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?
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Deduce this result from the Prime Number Theorem
    Posted in the Number Theory Forum
    Replies: 3
    Last Post: October 8th 2010, 03:44 AM
  2. Maximum error in the Prime Number Theorem
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: July 13th 2010, 08:31 AM
  3. Prime Number Theorem
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: March 22nd 2009, 07:45 AM
  4. A question about proof of prime number theorem
    Posted in the Number Theory Forum
    Replies: 6
    Last Post: September 16th 2008, 10:14 AM
  5. prime number theorem
    Posted in the Number Theory Forum
    Replies: 9
    Last Post: March 12th 2008, 07:37 AM

Search Tags


/mathhelpforum @mathhelpforum