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

Re: [PrimeNumbers] Digest Number 3824

Expand Messages
  • Chris De Corte
    This reference only shows that I independently proved Mertens theorem (my formula 9). It still doesn t answer why we need to adjust the calculated
    Message 1 of 2 , Feb 24, 2014
      This reference only shows that I independently proved Mertens' theorem (my formula 9).

      It still doesn't 'answer' 'why' we need to adjust the calculated probability with this constant to get the prime count. As mentioned, I am worried about this giant wave of prime multiples that is building up with this factor. But maybe it is all self-regulary by adjusting the dx.

      Also, so far, I haven't seen anyone using the probability product in an integration (my formula 6), which is just the key to the accurate prime counting results.



      From: "primenumbers@yahoogroups.com" <primenumbers@yahoogroups.com>
      To: primenumbers@yahoogroups.com
      Sent: Monday, February 24, 2014 2:36 PM
      Subject: [PrimeNumbers] Digest Number 3824

      There is 1 message in this issue.

      Topics in this digest:

      1a. Re: Probabilistic approach to prime counting   
          From:  djbroadhurst


      Message
      ________________________________________________________________________
      1a. Re: Probabilistic approach to prime counting
          Posted by:  david.broadhurst@... djbroadhurst
          Date: Sun Feb 23, 2014 4:34 am ((PST))

      Chris de Corte wrote:

      > The question of why we had to correct our probabilities
      > with a factor alpha [i.e. exp(Euler)] from 2 onward remains
      > open. And to be honest, we can’t give a good mathematical answer.

      Answer:

      http://mathworld.wolfram.com/MertensTheorem.html

      F. Mertens,
      "Ein Beitrag zur analytischen Zahlentheorie",
      J. reine angew. Math. 78 (1874) 46--62.

      http://gdz.sub.uni-goettingen.de/en/dms/load/toc/?PPN=PPN243919689_0078

      David







      Messages in this topic (3)



      Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
      The Prime Pages : http://primes.utm.edu/



      ------------------------------------------------------------------------
      Yahoo Groups Links

      <*> To visit your group on the web, go to:
          http://groups.yahoo.com/group/primenumbers/

      <*> Your email settings:
          Digest Email  | Traditional

      <*> To change settings online go to:
          http://groups.yahoo.com/group/primenumbers/join
          (Yahoo! ID required)

      <*> To change settings via email:
          primenumbers-normal@yahoogroups.com
          primenumbers-fullfeatured@yahoogroups.com

      <*> To unsubscribe from this group, send an email to:
          primenumbers-unsubscribe@yahoogroups.com

      <*> Your use of Yahoo Groups is subject to:
          http://info.yahoo.com/legal/us/yahoo/utos/terms/

      ------------------------------------------------------------------------


    • djbroadhurst
      ... You proved nothing. Rather you made a faulty heuristic and compared it with some data on small primes. Your heuristic contained one very obvious mistake:
      Message 2 of 2 , Feb 24, 2014
        Chris De Corte wrote:

        > I independently proved Mertens' theorem

        You proved nothing. Rather you made a faulty heuristic
        and compared it with some data on small primes.

        Your heuristic contained one very obvious mistake:
        you sieved out primes less than x, while we
        know that it is sufficient to sieve out primes less
        than or equal to sqrt(x), to ensure that x is prime.

        By this means you lost an obvious factor of

        2 = log(x)/log(sqrt(x)).

        Had you included this, your "fugde factor"
        would have been the famous factor

        2/exp(Euler) =~ 1.123

        by which the sieve of Eratosthenes outperforms
        a random sieve modelled by a Mertens product.

        To prove this, simply combine Mertens' theorem (1874)
        and the prime number theorem of Hadamard and
        de la Vallée-Poussin (1896), neither of which
        were proven in your "document".

        The factor 2/exp(Euler) has been discussed several times on this list:

        https://groups.yahoo.com/neo/groups/primenumbers/conversations/messages/21565

        https://groups.yahoo.com/neo/groups/primenumbers/conversations/topics/20936

        Hans Riesel discusses it on pp 66-67 of his book, following Theorem 3.1:

        http://tinyurl.com/y9whej4

        > the sieve of Eratosthenes sieves out numbers
        > more efficiently than does a "random" sieve


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