another problem with Infinite Products

*To*: mathgroup at smc.vnet.net*Subject*: [mg65481] another problem with Infinite Products*From*: Roger Bagula <rlbagulatftn at yahoo.com>*Date*: Wed, 5 Apr 2006 06:55:21 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Gary Adamson a long time ago came up with a sort of ultra-zeta sum: A=Sum[1/n^n,{n,1,Infinity}]=1.291.. That is very close to 2- Sqrt[2]... One wonders if the Product: Product[1/(1-1/Prime[n]^Prime[n]),{n,1,Infinity}]-->A or is it? Product[1/(1-1/Prime[n]^n),{n,1,Infinity}]-->A It is clear that both these products should converge and I think that the second should be A on reconsideration. But Mathematica disagrees: A = Sum[1/n^n, {n, 1, Infinity}] N[%] 1.2912859970626636` Product[1/(1 - 1/Prime[n]^Prime[n]), {n, 1, Infinity}] N[%] 1.3850602852044895` Product[1/(1-1/Prime[n]^n),{n,1,Infinity}] N[%] 2.2691047868959395`

**Follow-Ups**:**Re: another problem with Infinite Products***From:*Daniel Lichtblau <danl@wolfram.com>