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Used for?

Many problems in math have no intended practical uses (although uses, funnily enough, tend to turn up in time.)

Knowing what it might be used for really doesn't usually offer any special insight into the nature of the problem, nor how to solve it.

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Say it this way then: why that particular equation? Something must have prompted the formulation of the question - what was it?

The questions are answered in the article. The equivalence is not supposed to be obvious even to other number theorists. Most of the article is devoted to explaining "why" it works since the actual proof is two paragraphs.

Don't know what you're talking about. The article is about unproved theorems. They are described as being simple or obvious, but there's nothing obvious about

Prove that for n > 1, divsum(n) < H_n + exp(H_n)log(H_n)

why exp log? That's the question.

I believe that hyperbovine was referring to the article linked by tzs that explains the relation.

http://arxiv.org/pdf/math/0008177v2

See the entire field of number theory for something that didn't have applications when it was conceived. Gauss, Euler and the other early number theorists were essentially just amusing themselves with stuff like the phi function. Then cryptography became a serious open problem, and boom.

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