Alexandra Florea, a student of Soundararajan, has a nice new paper up, which I heard about in a talk by Michael Rubinstein. She computes the average of
as f ranges over squarefree polynomials of large degree. If this were the value at 1 instead of the value at 1/2, this would be asking for the average number of points on the Jacobian of a hyperelliptic curve, and I could at least have some idea of where to start (probably with this paper of Erman and Wood.) And I guess you could probably get a good grasp on moments by imitating Granville-Soundararajan?
But I came here to talk about Florea’s result. What’s cool about it is that it has the a main term that matches existing conjectures in the number field case, but there is a second main term, whose size is about the cube root of the main term, before you get to fluctuations!
The only similar case I know is Roberts’ conjecture, now a theorem of Bhargava-Shankar-Tsimerman and Thorne-Taniguchi, which finds a similar secondary main term in the asymptotic for counting cubic fields. And when I say similar I really mean similar — e.g. in both cases the coefficient of the secondary term is some messy thing involving zeta functions evaluated at third-integers.
My student Yongqiang Zhao found a lovely geometric interpretation for the secondary term the Roberts conjecture. Is there some way to see what Florea’s secondary term “means” geometrically? Of course I’m stymied here by the fact that I don’t really know how to think about her counting problem geometrically in the first place.
Quomodocumque 
COOL. I will definitely be reading this, thanks for publicizing it!
I got to read the paper in at least a bit more detail. A nice piece of work! Regarding your question to me: “Is there any Shintani-type reason to expect that term, the way there is for Roberts conjecture?”
If there were, then I’m pretty sure it would factor through geometry and representation theory: the Shintani zeta function machinery (and likewise Bhargava, Shankar, and Tsimerman’s “slicing” approach) allows you to take geometric properties and turn them into statements about secondary poles. But if there is no geometry to begin with, then there is no real way to get started.
Along these lines, please allow me to advertise a forthcoming paper of Taniguchi’s, where he reproves many of the analytic properties of the (binary cubic form) Shintani zeta function in a very simple way which cleanly illustrates what is going on.
There are some formal similarities between Florea’s argument and Shintani zeta function arguments (more so to Shintani’s work, than to e.g. my paper with Taniguchi). Also Florea gets at the central value via the approximate functional equation — which means that she turns into her problem into a “more honest” counting problem, which I think could well have a geometric interpretation. So I wonder if there is one!
Unfortunately, I don’t think that anything in the theory of Shintani zeta functions will help you go looking for it.
[…] Florea (last seen on the blog here) gave a great seminar here last year about quadratic L-functions over function fields, which gave […]