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

Re: [PrimeNumbers] product of 1- prime reciprocals

Expand Messages
  • mikeoakes2@aol.com
    In a message dated 29/06/04 17:10:22 GMT Daylight Time, tmgulland@hotmail.com ... By Euler s product expansion, zeta(1) = {product: p prime, p =2} 1/(1-1/p) =
    Message 1 of 2 , Jun 29, 2004
      In a message dated 29/06/04 17:10:22 GMT Daylight Time, tmgulland@...
      writes:


      > If you take the full sequence of primes starting at 11, and for each
      > prime, p, obtain a value for 1-(1/p), and multiply each of the
      > values, in turn, by the product of all the other values for obtained
      > for the lower values of p, can it be shown that the product will
      > never diminish to 0.5?
      >

      By Euler's product expansion,
      zeta(1) = {product: p prime, p>=2} 1/(1-1/p) = {sum: n >= 1} 1/n = infinity

      Taking the reciprocal:
      {lim: X => infinity}{product: p prime, p>=11, p<=X} (1-1/p) = 0.

      So your product /will/ get < 0.5 as X gets big enough.

      I hope this answers your question - or were you perhaps asking whether the
      product actually ever /equals/ 0.5?

      -Mike Oakes


      [Non-text portions of this message have been removed]
    Your message has been successfully submitted and would be delivered to recipients shortly.