Riemann Hypothesis

So the Riemann Hypothesis is difficult to explain; I shall let Wikipedia try, in my stead. Understanding why it's inherently interesting is also well beyond me — my usual recreational reading is generally either set theory or algorithmics.

But nonetheless I was reading about it recently, and I ran across this, listed among a collection of alleged proofs of the Riemann Hypothesis, with the sole descriptor being the word “What?”

The first line with a bracketed explanatory statement, I confess I cannot apprehend at all: there is introduced into the summand a factor of n‍^2s-1 from, as near as I can tell, nowhere in particular. I am not actually sure that this is erroneous, however, as the subconclusion a few lines below‍ —

ζ(s) = 0 ⇒ ζ(1-s) = 0
— happens to be true; I may just be missing something.

The more blatant error is actually below that, at the second bracketed statement. The bracketed statement alone is cause for bogglement: that F(x) = F(y) ⇒ x = y "for some range". In general this is of course absurd: consider any constant function. I could actually see something like this being true for the zeta function, given its universality — it seems plausible that something like ‘for any given open set U ⊆ ℂ, (∀z∈U: ζ(z) = ζ(z+k)) ⇒ (k = 0)’ might be valid, and possibly even for some class of non-open sets that could qualify as a "range". However, this does nothing to rule out zeroes at points, which the zeroes probably are anyway.

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Edited 2008-12-18: corrected typo in proposition (∈ for ⊆).