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About an identity by Euler

  1. Feb 25, 2004 #1
    Euler proved that the infinite product

    1/1-p**(-s) with p running over all primes was equal to R(s) where R(s) is teh Riemman z function my question is if the product:

    1/1-exp(-sp) over all primes would be hte same as summing the series

    1+exp(-s)+exp(-2s)+..=1/1+exp(-s) or what would be the equality...thanks.
  2. jcsd
  3. Feb 25, 2004 #2

    matt grime

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    Almost certainly not.

    In the orginal case, imagine multiplying out some finite portion and seeing what happens.

    they key thing to note is that the primes and up multiplied together, and you can sort of see how you get all the integers out of it.

    in the second the primes just get added together because they are in the exponents, so expecting to get the identity you conjecture is extremely unlikey.
  4. Feb 25, 2004 #3
    and do you know where i could find the answer to know the product:

    1/1-exp(-sp) porduct over all primes ?..
  5. Feb 25, 2004 #4

    matt grime

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    no, I don't know any identity it satisfies. why on earth should it even have a nicer form than that? have you even bothered to multiply out the first few terms to see what it looks like?
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