*[Corrected, thanks – T.]*

Unfortunately there is no rigorous direct connection currently known between function fields and number fields, which is a pity for many reasons (for instance, the Riemann hypothesis is known for the former but not the latter). Nevertheless the analogies between the two are remarkably strong. I believe part of the motivation of trying to construct a theory of “the field of one element” is to try to create such a connection, but this program has not yet succeeded in this task.

]]>where is the multiplicity for each prime in , is represented (at least formally) by the generating function

Which is equivalent to the fundamental theorem of arithmetic.

It is interesting to observe that the substitution in this partition generating function, gives Euler’s product representation for . Is similar connection exists between partition generating functions for permutations and polynomials and corresponding zeta functions?

]]>Well, at least some people think it’s conceptual.

*[Good point; I added a remark to this effect. -T.]*