Results 1 to 6 of 6

Math Help - Infinitely many primes (Riemann Zeta)

  1. #1
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Infinitely many primes (Riemann Zeta)

    Does any one know of a link to the proof that there are infinitely many primes using the Riemann Zeta function or would be willing to produce it?

    Thanks.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Infinitely many primes (Riemann Zeta)

    One of the most 'wonderful' discoveries of the Swiss mathematician Leonhard Euler was the 'product'...

    \sum_{n} \frac{1}{n^{s}} = \prod_{p} \frac{1}{1-\frac{1}{p^{s}}}= \zeta(s) (1)

    Now if s tends to 1, then \zeta(s) tends to infinity... what's the consequence?...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Re: Infinitely many primes (Riemann Zeta)

    Quote Originally Posted by chisigma View Post
    One of the most 'wonderful' discoveries of the Swiss mathematician Leonhard Euler was the 'product'...

    \sum_{n} \frac{1}{n^{s}} = \prod_{p} \frac{1}{1-\frac{1}{p^{s}}}= \zeta(s) (1)

    Now if s tends to 1, then \zeta(s) tends to infinity... what's the consequence?...

    Kind regards

    \chi \sigma
    Well, we need the function to be equal to \frac{\pi^2}{6}

    And if the Riemann Zeta function goes to infinity, then the sum and product are divergent. So we need s = 2.

    Assume there are finitely many primes.

    Then

    \zeta(2)\in\mathbb{Q}

    But we know \zeta(2)\notin\mathbb{Q}

    Therefore, we have reached a contradiction. Correct?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Infinitely many primes (Riemann Zeta)

    Quote Originally Posted by dwsmith View Post
    Well, we need the function to be equal to \frac{\pi^2}{6}

    And if the Riemann Zeta function goes to infinity, then the sum and product are divergent. So we need s = 2.

    Assume there are finitely many primes.

    Then

    \zeta(2)\in\mathbb{Q}

    But we know \zeta(2)\notin\mathbb{Q}

    Therefore, we have reached a contradiction. Correct?
    Yes it's correct... but what I had in mind is that the 'infinite sum' \sum_{n=1}^{\infty} \frac{1}{n^{s}} when s=1 is the 'armonic sum' \sum_{n=1}^{\infty} \frac{1}{n} which diverges to + \infty. If the 'discovery' of Leonhard Euler is true, then the product \prod _{p} \frac{1}{1-\frac{1}{p}} diverges to + \infty and that means that the product \prod_{p} (1-\frac{1}{p}) diverges to 0. Bur none of the terms 1-\frac{1}{p} is zero, so that necessarly there are infinite factors , i.e. there are infinite primes...

    Kind regards

    \chi \sigma
    Last edited by chisigma; June 22nd 2011 at 01:03 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Mar 2010
    From
    Florida
    Posts
    3,093
    Thanks
    5

    Re: Infinitely many primes (Riemann Zeta)

    What is the product of \zeta(1) since the the sum is divergent?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5

    Re: Infinitely many primes (Riemann Zeta)

    Even if with non precise speaking is \prod_{p} (1-\frac{1}{p})=\frac{1}{\zeta(1)}=0...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Infinitely many primes
    Posted in the Number Theory Forum
    Replies: 11
    Last Post: June 4th 2010, 06:43 AM
  2. infinitely many primes of the form 6k + 5 and 6K + 1
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: October 22nd 2009, 06:25 AM
  3. Infinitely Many Primes
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: June 24th 2009, 07:12 PM
  4. Riemann Zeta function - Zeta(0) != Infinity ??
    Posted in the Calculus Forum
    Replies: 4
    Last Post: March 8th 2009, 01:50 AM
  5. infinitely many primes
    Posted in the Number Theory Forum
    Replies: 0
    Last Post: November 18th 2008, 08:07 AM

Search Tags


/mathhelpforum @mathhelpforum