Loading ...
Sorry, an error occurred while loading the content.
 

Re: product convergence

Expand Messages
  • djbroadhurst
    ... Now let x tend to infinity and I think that you will see that K/log(x)^2 vanishes, not so :-? David
    Message 1 of 8 , May 5, 2010
      --- In primenumbers@yahoogroups.com,
      Kermit Rose <kermit@...> wrote:

      > I thought I had proven that the product converged to zero.

      It does. I wrote:

      > prod(2<p<x, (p-2)/p) ~ K/log(x)^2
      > K = 0.832429065661945278030805943531465575045445318077417053240894...

      Now let x tend to infinity and I think that you
      will see that K/log(x)^2 vanishes, not so :-?

      David
    • djbroadhurst
      ... It began OK and then fizzled out. Here is an elementary proof. Proposition: The product prod(p 2, (p-2)/p) over primes p 2 vanishes. Proof: It suffices to
      Message 2 of 8 , May 8, 2010
        --- In primenumbers@yahoogroups.com,
        Kermit Rose <kermit@...> wrote:

        > what is wrong with my "Proof"

        It began OK and then fizzled out.
        Here is an elementary proof.

        Proposition: The product prod(p>2, (p-2)/p) over primes p>2 vanishes.

        Proof: It suffices to show that the contrary proposition is absurd.
        Suppose that prod(p>2, (p-2)/p) did not vanish.
        Then, by taking logs, we would conclude that
        sum(p>2, log(p) - log(p-2)) is finite. But
        log(p) - log(p-2) > 2/p. Thus
        sum(p>2, 1/p) would also be finite.
        Yet that is easily proven to be absurd:
        http://primes.utm.edu/infinity.shtml#punchline
        Hence the product prod(p>2, (p-2)/p) does indeed vanish.

        Comment: By the same argument, it follows that
        the product prod(p>x, (p-x)/p) over primes p > x
        vanishes for all real x > 0.

        David
      Your message has been successfully submitted and would be delivered to recipients shortly.