...paper

This work remained unpublished until 1822.

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...sums

Note. We have tacitly changed the interval to 28#28 for convenience.

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...1.

Recall, (m,n) = 1 means m and n are relatively prime.

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...integrable?

This question of course has been answered. The relevant theorem is this:Theorem. Let J be a closed interval. The function f(x) is R-integrable over J if and only if it is continuous almost everywhere-J. In the case that f is non negative, these conditions in turn are equivalent to the graph of f(x) being (Jordan) measurable.

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...spaces

To name just a few, there are the Lebesgue, Hardy, Lipschitz, Sobolev, Orlicz, Lorentz and Besov spaces. Each space plays its own unique and important role in some slightly different area of analysis.

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...analysis

And this is an entire area of mathematics in and of itself.

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