@JasperLoy I think his anger at someone's actions was just enough to overcome his self-ban from chat. Now that he has said his piece, I don't think he will back for a while.
@mick well, Ramanujan sums are related to the theta functions and theta functions are the great voodoo hauptmoduls of the modforms so no wonder there is an arithmetic connection.
I think what is more interesting in the sense of analytic number theory are Kloosterman sums, which is a generalization of Ramanujan sums.
The connection with the sum of square function with Ramanujan sums are probably related to the so-called identity $$\sum_{n \geq 0} r_2(n) \exp(i2\pi nz) = \theta^2(z)$$, $\theta$ being one of the thetas.
So it's since recently that there have been quite a hubbub about Fourier series of modular forms.
@skullpatrol Wiles found a proof, but Fermat is the one who made it popular. Popular consensus may change the name to Fermat-Wiles, but that will be seen.
@BalarkaSen imagine, if you will, that there are various families of constructions that attach themselves to finite fields. in case after case, there is a "degenerate object" that would be associated with F_1, except if one actually plugs the trivial ring into the construction one doesn't get anything meaningful. so this degenerate-object-F1 correspondence exists on some higher level, beyond our current definitions. hence the heightened abstraction necessary to make sense of this phenomena...
the number of m dimensional subspaces of $\Bbb F_q^n$ is the q-binomial $\binom{n}{m}_q$. if one naively plugs in $q=1$, this formula counts the $m$-subsets of an $n$-element set, so that's what a space over F_1 should be
sure, one example shouldn't be too convincing (although that's a good one because it comes with a formula and plugging in q=1). but when it's example after example and expert intuition points the way...
that's a sign there is something going on behind the scenes
I think it was more of a cumulative thing. I'm not aware of the current state of research, but they've no doubt covered some ground in actually characterizing a definition or a set-up. search for it on arxiv.
there was one really cool determinant formula for L functions I saw on there once
I am not very interested in what connection there might be with Ford circles and the upper half plane under the action of J anyway, but the tiling with Farey sequence is something my professor told me about.
If I recall correctly, the tiling of the hyperbolic plane with Farey sequence is a powerful tool in diophantine approximations.
It gives you some kind of continued fraction (what it is, I forgot) that is used to derive some lower bounds.
I can't find the reference he sent me.
Oh well. I quickly jump from one thing to another. I should really go back concentrating about my commutative algebra.
