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.

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