15 Jan
2012
15 Jan
'12
4:23 a.m.
From: "Adam P. Goucher" <apgoucher@gmx.com>
Consider the infinite product:
(24/16) * (48/64) * (120/144) * ... * ((p^2 - 1)/(16*[p/4]^2)) * ...
In Sloane notation, this is:
Product (n >= 1) of (A061066(n+1) / (2*A024698(n+1)^2))
I've just realised that it has a much simpler representation:
Product (p >= 5) of:
(p+1)/(p-1) iff p = 4k + 1 (p-1)/(p+1) iff p = 4k - 1
How about: Product[p >= 5] (p+Chi4(p))/(p-Chi4(p)) Where Chi4() is the character mod 4 with Chi4(3)=-1, or equivalently the Legendre symbol (-1|p). Phil