## 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….

## Reader poll: how many times have you worn a necktie?

This Fermi question is probably easy for some people:  I’m guessing there are some women for whom the answer is zero (am I right?), and some men whose necktie-wearing is dominated by “every day at work, 5 days a week, 50 days a year.”  For me it’s much harder.  I guess I’d say — three or four times a year, as an adult?  More when I was younger and went to more weddings.  I’m gonna estimate I’ve put on a tie 150 times in my life.

Followup question:  how many ties do you own?  I think I probably have about 8, but 5 are in the “will never wear again” category, and one of the 3 “sometimes wear” is the American flag tie I only wear on July 4.

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## Why did Terrill Thomas die of thirst?

Nobody decided to kill Terrill Thomas.  He kept flooding his cell at the Milwaukee County Jail and making a mess so they just turned off the water to his cell.  Then they left it off until he was dead.  It took six days.  Fellow inmates say he was calling out for water.  Corrections officers say they checked in on Thomas every half hour, and “he had made some type of noise or movement” every time.  Until the last time, when he didn’t make any type of noise or movement because he’d died of thirst.

How did this happen?  It didn’t happen because David Clarke — the sheriff of Milwaukee County, and a top candidate to lead the Department of Homeland Security in the Trump administration — wanted to kill a prisoner in the most agonizing way imaginable.  What kind of psycho would want that?  I don’t think Sheriff Clarke wanted to kill a newborn baby either.  The baby was the fourth person to die in the jail since April.

These people died because nobody really seems to care what happens in the Milwaukee County Jail.  The medical services there are run by Armor Correctional Health Services, a company which oversees healthcare for 40,000 inmates in 8 states.  What are you saying about your priorities if you call your health care company “Armor”?

Armor’s glassdoor page doesn’t make it sound like a great place to work.  One employee writes:   “Stop being bean counters and start listening to your employees. We are asked to do too much: too many patients, too many intakes, not nearly enough staff to be in compliance with your own rules!”  Armor was sued by the state of New York this year over 12 inmates who died in the Nassau County Jail, including Daniel Pantera, who died of hypothermia in solitary confinement; they settled the suit last month for $350,000 and are barred for three years from bidding for contracts in the state. Armor does get a very nice endorsement, though, from Palm Beach County Sheriff Ric Bradshaw, who says right on the front of their website, “Armor stands out as an exemplary model of what partnership in correctional health should look like.” Bradshaw’s department was held liable this year for$22.4m in damages to Dontrell Stephens, and this summer settled for $550,000 against a former U.S. Marshall who said he was roughed up by deputies after stopping to help the victims of a traffic accident. Here in Wisconsin, Armor’s performance is overseen by Ronald Shansky, a court-appointed monitor and the first president of the Society of Correctional Physicians. Some of what Shansky has to say, based on his visit to Milwaukee County correctional facilities last month: As for the deaths: Shansky also, I should say, has a lot of praise for some staff members at the jail, characterizing them as devoted to their jobs and patients and doing the best they can under strained circumstances. And I believe that’s true. Again: the doctors of Armor didn’t want Terrill Thomas to spend six days dying of thirst. Neither did the CEO of Armor. Neither did David Clarke. But it happened. And everyone participated in creating the circumstances under which it happened, and under which it’s likely to happen again: public services outsourced to companies without the staff or resources to do the job right. It starts with jails. But it goes on to schools, to parking, to Medicare, to policing, to the maintenance of our bridges and roads. You’ll hear people say those services should be run like businesses. We can see in Milwaukee County what that looks like. Does it look good? Tagged , , , ## Call for nominations for the Chern Medal This is a guest post by Caroline Series. The Chern Medal is a relatively new prize, awarded once every four years jointly by the IMU and the Chern Medal Foundation (CMF) to an individual whose accomplishments warrant the highest level of recognition for outstanding achievements in the field of mathematics. Funded by the CMF, the Medalist receives a cash prize of US$ 250,000. In addition, each Medalist may nominate one or more organizations to receive funding totalling US\$ 250,000, for the support of research, education, or other outreach programs in the field of mathematics.

Professor Chern devoted his life to mathematics, both in active research and education, and in nurturing the field whenever the opportunity arose. He obtained fundamental results in all the major aspects of modern geometry and founded the area of global differential geometry. Chern exhibited keen aesthetic tastes in his selection of problems, and the breadth of his work deepened the connections of geometry with different areas of mathematics. He was also generous during his lifetime in his personal support of the field.

Nominations should be sent to the Prize Committee Chair: Caroline Series, email: chair(at)chern18.mathunion.org by 31st December 2016. Further details and nomination guidelines for this and the other IMU prizes can be found here.  Note that previous winners of other IMU prizes, such as the Fields Medal, are not eligible for consideration.

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## Double standards in baby names

People love to make fun of George Foreman because he named all his sons George Foreman.  But former Secretary of State Lawrence Eagleburger named all his sons Lawrence Eagleburger and nobody raises a peep!  There’s no justice.

## How the popular vote matters

Hillary Clinton lost the presidential election, but won the popular vote by 2% over Donald Trump, roughly in line with the popular margin George W. Bush enjoyed in his 2004 re-election campaign against John Kerry.

Does that matter?

As many of my conservative friends have pointed out, it doesn’t matter at all as far as who should be President.  Contests have rules.  If you don’t win the most games in the AL East, you don’t win the division; doesn’t matter if you scored more runs than the other teams and allowed fewer.  That’s not how we decide who wins.

It doesn’t matter; but it does matter.  If you actually want to know not just which team won the division, but which team is better at baseball, you do want to keep track of runs scored and runs allowed.  Same thing if you want to make predictions about which team will win the division next time.  Or to give good advice as to whether a team needs to reshape its whole strategy or is best off sticking with its current approach.

My conservative friends also like to point out that the United States is a republic, not a democracy.  They’re right about that too.  Our electoral system, by design, will sometimes choose as President someone the American people don’t prefer and whose promised policies most of us don’t want enacted.  With a little effort you can even come up with a story that makes sense of this:  at some moments, you imagine, you need a President determined to protect the interests of the more vulnerable parts of America against the crude rule of the majority, who has a clearly articulated political vision that doesn’t sway with the gusts of public opinion, who fundamentally doesn’t mind being disagreed with, even disrespected, by the majority of the people he serves.

I don’t think that’s the guy we got.

## A Man Who Suddenly Fell Over

“A Man Who Suddenly Fell Over,” by Michael Andrews (1952)

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## The greatest Cub/Indian

Congratulations to the Cubs, the Indians, and their fanbases, one of which will enjoy a long-awaited championship!

Now here’s the question.  Which player in baseball history was the best combined Cub/Indian?  My methodology, as it was last year, is to draw the top 200 position players and pitchers from each team by Wins Above Replacement, using the Baseball Reference Play Index.  Then I find the players with the highest value of

(WAR for team 1 * WAR for team 2)

Now I have to admit I couldn’t actually think of a player who played for both the Cubs and the Indians!  And this was borne out by the Play Index results:  there were only five position players and no pitchers who ranked in the top 200 all-time contributors to each team.  Pretty surprising, considering how long both teams have been around!  And here are your top five Cub/Indians:

1.  Riggs Stephenson (193.6)
2.  Andre Thornton (98.8)
3.  Jose Cardenal (47.4)
4.  Mel Hall (9.0)
5.  Mitch Webster (7.8)

I almost wonder whether I did something wrong here.  There was so much more overlap last year between the Royals and the Mets!  But until you tell me otherwise, it’s the Riggs Stephenson Series.

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## Math!

I really like talking with AB about arithmetic and her strategies for doing problems.  All this Common Core stuff about breaking up into hundreds and tens and ones that people like to make fun of?  That’s how she does things.  She can describe her process a lot more articulately than most grownups can, because it’s less automatic for her.  I learn a lot about how to teach math by watching her learn math.

## Orioles postmortem 2016

What is there to say?  Should Showalter have used Britton?  Probably.  When?  Probably when O’Day came in, when the Orioles desperately needed a double play to keep the game tied.  (But O’Day got the double play ball anyway.)  Barring that, you bring Britton in to start the 11th, I think, because Britton doesn’t give up home runs and you’ve got the home run guys coming up.  But what difference does it make?  The Orioles weren’t hitting, not off Liriano, not off anybody.  Jimenez would have been come in to pitch the 12th or 13th anyway.  The real mistake was pinch hitting Reimold for Kim.  Why?  Kim is the only guy on the team who gets on base.  Maybe he walks and Machado comes up and you have an actual chance.  Reimold is a bad defender, too; his misplay in the 11th, letting Devon Travis get to third, could have been decisive if Encarnacion had hit a single instead of a home run.

I said on Twitter it reminded me of the last game of the 1997 ALCS, but when I think it over, this one was a lot less heartbreaking.  In that game, Mike Mussina delivered one of the best Orioles playoff starts of my lifetime, and we wasted it.  Ten hits and five walks and we couldn’t push one run across.

And here’s the thing.  That 1997 team was the best Orioles squad in 15 years, and you had the real sense it was a one-shot deal.  The next year we were back to losing.  The 2016 team is probably the third-best in the last five years, and the main contributors will all be back next year.  It’s been a big adjustment, rooting for a team that’s consistently good, but my ability to absorb this loss makes me think it’s finally starting to sink in.

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