30 Mar
2008
30 Mar
'08
3:18 p.m.
I wrote: [Steve Witham:]
f(i) = i'th prime (1/2) (2/3) (4/5) (6/7) (10/11) (12/13) ... = 0
(The fraction of counting numbers that aren't divisible by any prime.) Does that equation have a name?
[me:]
product (1-p^-s) = 1 / sum (n^-s) = 1/zeta(s) and indeed the zeta function has a pole at 0 so the product is 0.
Ahem. (1) I meant "at 1", not "at 0", and (2) it's kinda bogus to be trying to define zeta(s) by that formula when Re(s) <= 1. (Though the singularity at s=1 is deducible from the behaviour for, say, s real and > 1, where the stuff above is unproblematic.) Thanks to Dan Asimov for pointing out my screwups off-list :-). -- g