Actually, proving the order a square when finite would seem to be a candidate. But he likes his jokes.

The tails of such sums are easily bounded by integrals.

Off-topic, but that poll you link to is a hilarious beauty contest, no more. Kripke one place behind Hegel, and one ahead of Nietzsche? Come now. And who's idea was "200 years" so that Kant is excluded?

What Gauss said is typical old-style Ciceronian rhetoric, and I don't suppose anyone at the time would have taken it very seriously.

@ConfuseD: Could you edit the question to state the new version?

@Igor: Unless you meant a Hilbert space is always "virtual".

@Igor: According to a well-known reference site, "A Hilbert space H is a real or complex inner product space that is also a complete metric space with respect to the distance function induced by the inner product."

Abstractly you're intersecting a sphere in a real Hilbert space with a hyperplane. The hyperplane is a closed affine-linear subspace, by Cauchy-Schwarz. So the geometry isn't too bad. You seem to want some parametric information on the intersection. "Combining the functions" to you seems to mean working in the algebra they generate. Which is in the territory of the Stone-Weierstrass theorem, though that's for the complex (uniform) algebra, applying to continuous functions. There must clues in the functional analysis, but your formulation suggests you want a simple answer.

Shrug. We need to agree to differ here.

"This correspondence is closed". The topic turned out to be divisive (some upvotes while others decided it wasn't a real question). I was learning while the "Red book" was still red: the "real question" would be who is Mumford's successor, with that way of connecting Bures-sur-Yvette with Harvard, geometry with algebra, classical with modern.

Yes, see en.wikipedia.org/wiki/Local_zeta-function, but in what form are you seeking an answer?

@Emerton, indeed, this is one type of answer I'd expect: specialists can write texts in their area of expertise. But how about taking the point of view of the consumer, not the producer? In other fields, in fact, the situation would be considered fairly scandalous.