Quite hard to do justice to all that: see "Why are topological ideas so important in arithmetic?" and comments.

Ah, Robinned again.

See en.wikipedia.org/wiki/Diophantine_geometry for more accurate and neutral perspective (I hope). Gauss is alleged to have said about FLT that he thought it should be developed in a broader context. Kummer initiated a theory, and, as you say, this special equation had a broad influence. General methods for Diophantine equations really started with Thue, and I wouldn't call Diophantine approximation "topological"; post-Vojta there is a new view? FLT's solution I don't understand, but deformation of Galois representations are bounding Selmer groups? There is topology on one side there.

You realise that Alvarez adds 1/5x6, 1/6x7, 1/7x8, 1/10x11, 1/11x12 etc.?

I have to say that I'm not going to type more mathematical notation here, given your reaction above. It is a bit ironic that you ask for an explanation in mathematical language. It is not clear to me whether this is a challenge problem to which you already know the answer, but I'm not highly motivated yet (whether it is or not). You have ignored the request I made to formulate the series in sigma notation: "..." can waste a lot of time, as the discussion shows.

OK, in this forum, please don't use "compsci" notation for multiplication, and please do use sigma notation. We can all write 1/6 = 1/2 - 1/3 here, so could you verify that my a(k) is what you want?

It's just irritating to conceal the partial fraction decomposition, though.

I’m now considering Ʃ a(k)/k where a(k) = 0 when k is a square, and otherwise a(k) = -1^b(k) with b(k) = k + [square root(k)], [ ] being integral part. This is meant to be equivalent to the OP's intended series, up to some factor of plus or minus 2 or something.