## Hast and Matei, “Moments of arithmetic functions in short intervals”

Two of my students, Daniel Hast and Vlad Matei, have an awesome new paper, and here I am to tell you about it!

A couple of years ago at AIM I saw Jon Keating talk about this charming paper by him and Ze’ev Rudnick.  Here’s the idea.  Let f be an arithmetic function: in that particular paper, it’s the von Mangoldt function, but you can ask the same question (and they do) for Möbius and many others.

Now we know the von Mangoldt function is 1 on average.  To be more precise: in a suitably long interval ($[X,X+X^{1/2 + \epsilon}]$ is long enough under Riemann) the average of von Mangoldt is always close to 1.  But the average over a short interval can vary.  You can think of the sum of von Mangoldt over  $[x,x+H]$, with H = x^d,  as a function f(x) which has mean 1 but which for d < 1/2 need not be concentrated at 1.  Can we understand how much it varies?  For a start, can we compute its variance as x ranges from 1 to X?This is the subject of a conjecture of Goldston and Montgomery.  Keating and Rudnick don’t prove that conjecture in its original form; rather, they study the problem transposed into the context of the polynomial ring F_q[t].  Here, the analogue of archimedean absolute value is the absolute value

$|f| = q^{\deg f}$

so an interval of size q^h is the set of f such that deg(f-f_0) < q^h for some polynomial f_0.

So you can take the monic polynomials of degree n, split that up into q^{n-h} intervals of size q^h, and sum f over each interval, and take the variance of all these sums.  Call this V_f(n,h).  What Keating and Rudnick show is that

$\lim_{q \rightarrow \infty} q^{-(h+1)} V(n,h) = n - h - 2$.

This is not quite the analogue of the Goldston-Montgomery conjecture; that would be the limit as n,h grow with q fixed.  That, for now, seems out of reach.  Keating and Rudnick’s argument goes through the Katz equidistribution theorems (plus some rather hairy integration over groups) and the nature of those equidistribution theorems — like the Weil bounds from which they ultimately derive — is to give you control as q gets large with everything else fixed (or at least growing very slo-o-o-o-o-wly.)  Generally speaking, a large-q result like this reflects knowledge of the top cohomology group, while getting a fixed-q result requires some control of all the cohomology groups, or at least all the cohomology groups in a large range.

Now for Hast and Matei’s paper.  Their observation is that the variance of the von Mangoldt function can actually be studied algebro-geometrically without swinging the Katz hammer.  Namely:  there’s a variety X_{2,n,h} which parametrizes pairs (f_1,f_2) of monic degree-n polynomials whose difference has degree less than h, together with an ordering of the roots of each polynomial.  X_{2,n,h} carries an action of S_n x S_n by permuting the roots.  Write Y_{2,n,h} for the quotient by this action; that’s just the space of pairs of polynomials in the same h-interval.  Now the variance Keating and Rudnick ask about is more or less

$\sum_{(f_1, f_2) \in Y_{2,n,h}(\mathbf{F}_q)} \Lambda(f_1) \Lambda(f_2)$

where $\Lambda$ is the von Mangoldt function.  But note that $\Lambda(f_i)$ is completely determined by the factorization of $f_i$; this being the case, we can use Grothendieck-Lefschetz to express the sum above in terms of the Frobenius traces on the groups

$H^i(X_{2,n,h},\mathbf{Q}_\ell) \otimes_{\mathbf{Q}_\ell[S_n \times S_n]} V_\Lambda$

where $V_\Lambda$ is a representation of $S_n \times S_n$ keeping track of the function $\Lambda$.  (This move is pretty standard and is the kind of thing that happens all over the place in my paper with Church and Farb about point-counting and representation stability, in section 2.2 particularly)

When the smoke clears, the behavior of the variance V(n,h) as q gets large is controlled by the top “interesting” cohomology group of X_{2,n,h}.  Now X_{2,n,h} is a complete intersection, so you might think its interesting cohomology is all in the middle.  But no — it’s singular, so you have to be more careful.  Hast and Matei carry out a careful analysis of the singular locus of X_{2,n,h}, and use this to show that the cohomology groups that vanish in a large range.  Outside that range, Weil bounds give an upper bound on the trace of Frobenius.  In the end they get

$V(n,h) = O(q^{h+1})$.

In other words, they get the order of growth from Keating-Rudnick but not the constant term, and they get it without invoking all the machinery of Katz.  What’s more, their argument has nothing to do with von Mangoldt; it applies to essentially any function of f that only depends on the degrees and multiplicities of the irreducible factors.

What would be really great is to understand that top cohomology group H as an S_n x S_n – representation.  That’s what you’d need in order to get that n-h-2 from Keating-Rudnick; you could just compute it as the inner product of H with $V_\Lambda$.  You want the variance of a different arithmetic function, you pair H with a different representation.  H has all the answers.  But neither they nor I could see how to compute H.

Then came Brad Rodgers.  Two months ago, he posted a preprint which gets the constant term for the variance of any arithmetic function in short intervals.  His argument, like Keating-Rudnick, goes through Katz equidistribution.  This is the same information we would have gotten from knowing H.  And it turns out that Hast and Matei can actually provably recover H from Rodgers’ result; the point is that the power of q Rodgers get can only arise from H, because all the other cohomology groups of high enough weight are the ones Hast and Matei already showed are zero.

So in the end they find

$H = \oplus_\lambda V_\lambda \boxtimes V_\lambda$

where $\lambda$ ranges over all partitions of n whose top row has length at most n-h-2.

I don’t think I’ve ever seen this kind of representation come up before — is it familiar to anyone?

Anyway:  what I like so much about this new development is that it runs contrary to the main current in this subject, in which you prove theorems in topology or algebraic geometry and use them to solve counting problems in arithmetic statistics over function fields.  Here, the arrow goes the other way; from Rodgers’s counting theorem, they get a computation of a cohomology group which I can’t see any way to get at by algebraic geometry.  That’s cool!  The other example I know of the arrow going this direction is this beautiful paper of Browning and Vishe, in which they use the circle method over function fields to prove the irreducibility of spaces of rational curves on low-degree hypersurfaces.  I should blog about that paper too!  But this is already getting long….

## 7 thoughts on “Hast and Matei, “Moments of arithmetic functions in short intervals””

1. David Speyer says:

As I’m sure you know, $\bigoplus V_{\lambda} \boxtimes V_{\lambda}$ is the group algebra $\mathbb{C}[S_n]$ as an $S_n \times S_n$ rep. Let $\mathbb{C}[x_{ij}]$ be a polynomial ring in $n^2$ variables thought of as the entries of an $n \times n$ matrix, and think of $\mathbb{C}[S_n]$ as the representation on monomials of the form $x_{1 \sigma(1)} x_{2 \sigma(2)} \cdots x_{n \sigma(n)}$. Then I believe $\bigoplus_{\lambda_1 \leq r} V_{\lambda'} \boxtimes V_{\lambda'}$ (where $\lambda'$ is the transpose partition) is the quotient of this representation where we restrict these monomials to matrices of rank $\leq r$.

For example, if $r=n-1$, there is one partition which violates the condition, namely $(n)$. Its transpose is $(111 \cdots 1)$, corresponding to the sign representation. Matrices of rank $\leq n-1$ are cut out by $\det =0$, which is a single linear relation in the monomials in question. So restricting to these matrices quotients by a one dimensional subrepresentation of $\mathbb{C}[S_n]$, which is $V_{11 \cdots 1} \boxtimes V_{11 \cdots 1}$.

I was reading this fact recently; I’ll find you a reference when I have more time. What would you want to know about this representation?

2. JSE says:

Whoa, I didn’t know this! I’m not sure what facts I (or, more precisely, Daniel and Vlad) want to know about this representation; I just found it striking and wondered whether it was something that naturally shows up other places. Sounds like yes.

3. NDE says:

[typos, or rather TeXos: “infty” should be “\infty” in the 2nd displayed equation; stray subscript “)” in the fourth; and probably “deg” should be “\deg” in the 1st]

4. NDE says:

[TeXos FiXed, thAnX]

5. […] was about sums of arithmetic functions on short intervals in function fields.  As I was saying when I blogged about Hast and Matei’s paper, a short interval in F_q[t] means:  the set of monic degree-n polynomials P such […]

6. Will Sawin says:

This representation also arises as the ring of GL_r-equivariant endomorphisms of V^{\otimes n}. One can establish this directly as a consequence of Katz’s monodromy computations, without reference to point-counting. (Although the monodromy computations are proven using point-counting on some simple special cases of this variety!).

In particular, one can see that there are n! “obvious” cohomology classes obtained from the locus where f_1 = f_n, which splits into n! different subvarieties depending on the ordering of the roots. I think one can see that the classes of these subvarieties generate this space, by corresponding to the to the endomorphisms of V^{\otimes n} given by elements of S_n.

7. […] New paper up!  With my Ph.D. student Daniel Hast (last seen on the blog here.) […]