TITLE: Applications of algebraic geometry over a field with one element
QUESTION [23 upvotes]: I would like to understand at least one of the several existing approaches to algebraic geometry over $\mathbb{F}_1$ (the field with one element). Is there an example of an "interesting" theorem that can be formulated purely in the language of ordinary schemes, but which can be proved using algebraic geometry over $\mathbb{F}_1$?
Of course, the interpretation of the word "interesting" is entirely up to your own taste. An example in which the theorem cannot be proved using "classical" methods would be most desirable, but examples where (one of) the theories of schemes over $\mathbb{F}_1$ gives an alternative proof of an already known result would also be very much appreciated.
[On a related note, perhaps there should be a tag "naive-question" for situations like this one.]

REPLY [11 votes]: To elaborate on the possible connection with the Riemann hypothesis:
One of the parts of the Weil conjectures (which were proved by Deligne, and which also follow from Grothendieck's "standard conjectures") states:

If $X$ is a smooth projective variety of dimension $n$ over $\mathbb{F}_q$, then its zeta function $Z_X(t)$ has the form $$\frac{P_1(t) P_3(t) \cdots P_{2n-1}(t)}{P_0(t)P_2(t) \cdots P_{2n}(t)},$$ where each $P_i(t)$ has the form $\prod_j (1-\alpha_{ij}t)$, and such that each $\alpha_{ij}$ is an algebraic integer such that $|\alpha_{ij}| = q^{i/2}$.

The zeta function $Z_X(t)$ of a variety $X$ is defined to be $\exp(\sum N_r t^r/r)$, where $N_r$ is the number of $\mathbb{F}_{q^r}$-points of $X$.
Write $\zeta_X(s) = Z_X(q^{-s})$. Then the above statement implies that the zeros and poles of $\zeta_X(s)$ lie on the lines given by $\mathfrak{Re}(s) = i/2$ with $i=1, \dots, 2n$. The function $\zeta_X(s)$ is also equal to $$\prod_{x \in X} \frac{1}{1-|k(x)|^{-s}}$$ where $|k(x)|$ is the order of the residue field of $x$ (which is always finite).
Now if $X = \operatorname{Spec} \mathbb{Z}$ were somehow a smooth projective variety of dimension 1 over $\operatorname{Spec} \mathbb{F}_1$, then $\zeta_X(s)$ would be the Riemann zeta function, and...