```Date: Apr 12, 2009 3:29 PM
Author: James A. Landau
Subject: Prime numbers and pi

On Monday, 30 Mar 2009 04:12:42 EDT Franz Gnaedinger <discussions@MATHFORUM.ORG> wrote:<begin quote><snip>A fascinating oscillation occurs in Euler's infinite multiplication that connectspi with the primes (the primes constitute the numerators while the denominators oscillate around the primes, avoiding four and multiples of four):pi = 2 x 3/2 x 5/6 x 7/6 x 11/10 x 13/14 x 15/14 ...<end quote>Thank you for showing Euler's infinite product, which I knew existed but never managed to find a copy of before.This product is a variation on Wallis's infinite product for pipi   2x2   4x4   6x6   8x8-- = --- x --- x --- x --- x ... 2   1x3   3x5   5x7   7x9   Looking at Wallis's product we can see that the connection with primes is rather tenuous, resulting from an infinite multiplication of odd and even integers with cancellation.Two other tenuous connections between primes and pi are:1.  the Riemann Zeta function is connected with the distribution of primes.    Zeta (2) = pi^2/6    Zeta (4) = pi^4/90    Zeta (6) = pi^6/945   etc.(these values were found by Euler)2.  the Fourier transform 1/sqrt(2 pi)integral (-infinity to +infinity) f(t)exp(xit) dtand more specifically the Fast Fourier Transform, one version of which uses the Chinese Remainder Theorem.   - James A. Landau_____________________________________________________________Netscape.  Just the Net You Need.
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