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product convergence

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  • tsr21
    Hi all, A correspondent has asked me about the infinite product (p-2)/p where p is a prime from 5 to infinity. So the first few terms are 3/5, 5/7, 9/11,
    Message 1 of 8 , May 3, 2010
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      Hi all,

      A correspondent has asked me about the infinite product (p-2)/p where p is a prime from 5 to infinity. So the first few terms are
      3/5, 5/7, 9/11, 11/13, 15/17 etc.

      Clearly the terms converge to 1, and the product - 3/5, then 3/7, then 27/77, then 27/91, etc - also converges. But does the product converge to zero, or to a small but finite number?

      I have been unable to answer him definitively. Can anyone else here do so?

      Tim
    • Andrey Kulsha
      ... 27/77, then 27/91, etc - also converges. But does the product converge to zero, or to a small but finite number? ... It converges to zero as well as the
      Message 2 of 8 , May 3, 2010
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        > Clearly the terms converge to 1, and the product - 3/5, then 3/7, then
        27/77, then 27/91, etc - also converges. But does the product converge to
        zero, or to a small but finite number?
        >
        > I have been unable to answer him definitively. Can anyone else here do so?

        It converges to zero as well as the product of (p-1)/p.

        Best regards,

        Andrey
      • tsr21
        Thanks Andrey. Is there a simple proof of this? Tim
        Message 3 of 8 , May 3, 2010
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          Thanks Andrey. Is there a simple proof of this?

          Tim

          --- In primenumbers@yahoogroups.com, Andrey Kulsha <Andrey_601@...> wrote:
          >
          > > Clearly the terms converge to 1, and the product - 3/5, then 3/7, then
          > 27/77, then 27/91, etc - also converges. But does the product converge to
          > zero, or to a small but finite number?
          > >
          > > I have been unable to answer him definitively. Can anyone else here do so?
          >
          > It converges to zero as well as the product of (p-1)/p.
          >
          > Best regards,
          >
          > Andrey
          >
        • djbroadhurst
          ... See Hardy and Wright Sections 22.7 and 22.8 for the Mertens product prod(p
          Message 4 of 8 , May 4, 2010
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            --- In primenumbers@yahoogroups.com,
            "tsr21" <timro21@...> wrote:

            > Is there a simple proof of this?

            See Hardy and Wright Sections 22.7 and 22.8
            for the Mertens product

            prod(p<x, (p-1)/p) ~ exp(-Euler)/log(x)

            where the product runs over primes.

            As Andrey remarked, your product tends to zero faster:

            prod(2<p<x, (p-2)/p) = O(1/log(x)^2)

            David
          • djbroadhurst
            ... So let s work out the constant, say K, in prod(2
            Message 5 of 8 , May 4, 2010
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              --- In primenumbers@yahoogroups.com,
              "djbroadhurst" <d.broadhurst@...> wrote:

              > prod(2<p<x, (p-2)/p) = O(1/log(x)^2)

              So let's work out the constant, say K, in

              prod(2<p<x, (p-2)/p) ~ K/log(x)^2

              We should use the twin-prime constant

              C2 = prod (2<p, p*(p-2)/(p-1)^2) = 0.6601618158...

              and then use the square of Mertens' formula,
              remembering that the latter includes p = 2.

              K = C2*(exp(-Euler)*2)^2 =
              0.832429065661945278030805943531465575045445318077417053240894...

              Sanity check:

              default(primelimit,10^8);
              \p5
              P=1.;x=10^8;forprime(p=3,x,P*=1-2/p);print(P*log(x)^2);

              0.83242

              Looks OK to me...

              David
            • Kermit Rose
              ... Hmm.... And I thought I had proven that the product converged to zero. David, what is wrong with my Proof below which I had already sent to Tim? Kermit
              Message 6 of 8 , May 5, 2010
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                primenumbers@yahoogroups.com wrote:
                >
                >
                >
                > Messages in this topic (5)
                > ________________________________________________________________________
                > 1b. Re: product convergence
                > Posted by: "djbroadhurst" d.broadhurst@... djbroadhurst
                > Date: Tue May 4, 2010 1:17 pm ((PDT))
                >
                >
                >
                > --- In primenumbers@yahoogroups.com,
                > "djbroadhurst" <d.broadhurst@...> wrote:
                >
                >
                >> prod(2<p<x, (p-2)/p) = O(1/log(x)^2)
                >>
                >
                > So let's work out the constant, say K, in
                >
                > prod(2<p<x, (p-2)/p) ~ K/log(x)^2
                >
                > We should use the twin-prime constant
                >
                > C2 = prod (2<p, p*(p-2)/(p-1)^2) = 0.6601618158...
                >
                > and then use the square of Mertens' formula,
                > remembering that the latter includes p = 2.
                >
                > K = C2*(exp(-Euler)*2)^2 =
                > 0.832429065661945278030805943531465575045445318077417053240894...
                >
                > Sanity check:
                >
                > default(primelimit,10^8);
                > \p5
                > P=1.;x=10^8;forprime(p=3,x,P*=1-2/p);print(P*log(x)^2);
                >
                > 0.83242
                >
                > Looks OK to me...
                >
                > David
                >
                >
                >


                Hmm.... And I thought I had proven that the product converged to zero.

                David, what is wrong with my "Proof" below which I had already sent to Tim?

                Kermit


                > *****************************************************
                >
                >
                >
                > Hello Tim.
                >
                >
                >
                > http://en.wikipedia.org/wiki/Infinite_product
                >
                >
                >
                > The infinite product 3/5 5/7 9/11 ...
                >
                > converges to zero.
                >
                >
                >
                > would converge to a positive number between 0 and 1 only if
                >
                > the infinite product
                >
                > 5/3 7/5 11/9 ..... converged to a positive number > 1.
                >
                >
                >
                > 5/3 7/5 11/9 ..... = (1 + 2/3) (1 + 2/5) (1 + 2/9) (1 + 2/11) (1 +
                > 2/15) (1 + 2/17) ...
                >
                >
                > 1 + 2/3 + 2/5 + 2/9 + 2 / 11 + .... =< (1 + 2/3) (1 + 2/5) (1 + 2/9)
                > (1 + 2/11) (1 + 2/15) (1 + 2/17) ... =< exp( 2/3 + 2/5 + 2/9 + 2/11
                >
              • djbroadhurst
                ... Now let x tend to infinity and I think that you will see that K/log(x)^2 vanishes, not so :-? David
                Message 7 of 8 , May 5, 2010
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                  --- 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 8 of 8 , May 8, 2010
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                    --- 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
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