Is this product mentioned in any of the literature? I haven't been able to find it anywhere previous to Eric's email!

Tim

-----Original Message-----

From:

mikeoakes2@... [mailto:

mikeoakes2@...]

Sent: Sun 21/12/2003 8:07 PM

To:

primenumbers@yahoogroups.com
Cc:

Subject: Re: [PrimeNumbers] pi=(2/1)x(3/2)x(5/6)x(7/6)x(11/10)x(13/14)x(17/18)x(19/18)...

In a message dated 21/12/03 02:05:29 GMT Standard Time,

lh2072@...
writes:

> My guess would be

>

> pi =infiniteproduct(over all primes = 1 mod 4 of p/(p+1)) *

> infiniteproduct(over all other primes of p/(p-1))

>

> Next terms:

> 29/30 * 31/30 * 37/38...

>

You've got it!

And here is the proof of why the sum = pi.

Define the "character" chi(p) by:-

chi(p) = +1 if p is a prime = +1 mod 4

chi(p) = -1 if p is a prime = -1 mod 4

chi(p) = 0 for all other integers p > 0.

Eric's product is:-

(2/1)x(3/2)x(5/6)x(7/6)x(11/10)x(13/14)x(17/18)x(19/18)x(23/22)...

= 2/P, say

where P=(2/3)*(6/5)*(6/7)*(10/11)*(14/13)*...

= (1-1/3)*(1+1/5)*(1-1/7)*(1-1/11)*(1+1/13)*...

= product{p odd prime}(1+chi(p)/p)

= product{p odd prime}(1-1/p^2) / product{p odd prime}(1-chi(p)/p)

= [(4/3)* product{p prime}(1-1/p^2)] / product{p odd prime}(1-chi(p)/p)

= [(4/3)/zeta(2)] * L(chi,1)

= [(4/3)/(pi^2/6)] * [pi/4] = 2/pi

where zeta(s) is the Riemann zeta function and L(chi,s) is the Dirichlet

L-function associated with chi().

Q.E.D.

A useful reference for the background to all this is:

http://mathworld.wolfram.com/DirichletL-Series.html
-Mike Oakes

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