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Representation of e in terms of primes

  1. Jan 30, 2014 #1
    We can represent π, in terms of primes by using Euler's product form of Riemann Zeta.
    For example ζ(2)=(π^2)/6= ∏ p^2/(p^2-1).

    Likewise, is there a representation of e that is obtained by using only prime numbers?
     
  2. jcsd
  3. Feb 4, 2014 #2
    I guess there is no such known representation?
     
  4. Feb 26, 2017 #3
    There are two expressions relating e to the prime number distribution:

    $$ e = \lim_{n\to \infty} n^{\pi(n)/n} $$

    and

    $$ e = \lim_{n\to \infty} (p_n \#)^{1/p_n} $$

    where ## \pi(n) ## is the prime counting function, ##p_n## is the n-th prime and ##p_n \# ## is the primorial of ##p_n ##. (see https://en.wikipedia.org/wiki/List_of_representations_of_e )

    Best wishes,

    DaTario
     
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