Okay, I'm confused on something that's probably rather basic but it's really annoying me so a little help would be appreciated.

I'm reading a book and it's going over how the Golden Key was arrived at by basically using the Sieve of Eratosthenes on the Riemann Zeta Function.

And it gets to this point

ζ(s)=1+1/2s+1/3s+1/4s+....

Then you multiply both sides by the fist expression and subtract the new expression by the first expression and you end up hypothetically with something like:

...(1-1/7s)(1-1/5s)(1-1/3s)(1-1/2s)ζ(s)=1

Well here’s the part that just isn’t computing to me, it’s saying that the next step that was taken was dividing the left side by the right side, fair enough if it ended up being ζ(s)=1/(the entire term that was on the left).

Instead, it says that after dividing the left side by the right side we end up with:

ζ(s)=[(1)/1-1/2s))*((1)/(1-1/3s)]...

This makes absolutely no sense to me, and I tried working it out myself by doing this:

(1/2)(1/2)(1/2) = a; where a obviously is equal to 1/8

and if you divide how it shows to divide, you end up with:

1= (.125/.5)x(.125/.5)x(.125/.5)

And this obviously isn’t right, so could someone please explain how this was arrived at correctly?