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>

A fascinating oscillation occurs in Euler's infinite multiplication that connects
pi 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 pi

pi 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) dt

and more specifically the Fast Fourier Transform, one version of which uses the Chinese Remainder Theorem.

- James A. Landau

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