Interesting points raised so far, I'm still investigating. I was thinking of the recursive procedure and the cache/remember option, but ran into the same issues already discussed.

(on an aside my computations of the coding you have provided seem to be taking twice as long on my computer, but that is probably just the computer)

Any further discussion and ideas are most welcome.

Is it possible that a while loop is the wrong approach altogether? can it be done another way faster? Using "if" clauses perhaps?

Code for Euclid (1D math unfortunatly):

euclid := proc (s, n)
  if 1 < s and 1 <= n then
    product(1/(1-ithprime(i)^(-s)), i = 1 .. n)
  end if
end proc;