E. Kowalski’s blog

Having just sent the title

Erdös-Kác, Rényi-Turán, Keating-Snaith, and Katz-Sarnak

for a lecture at a forthcoming conference, I was naturally led to wonder about the marvelous English expression name-dropping, and I resorted to the OED for information. I was surprised to see that the word is claimed to go back no further than the 1940′s; in fact the first quotation is for name-dropper:

1939 Los Angeles Times 17 Jan. 115/5 My pet aversion..is the name dropper, the type that is always saying: ‘Well,..when I had lunch with the P. of W., he said-’ I say to them: ‘The P. of What?’ ‘The P. of W., the Prince of Wales, of course,’ they say.

The first instance of name-dropping is in 1945; a nice quotation from 1999 is

1999 Times 16 July 24/7 Name-dropping is so vulgar, as I was telling the Queen last week.

Considering that the phenomenon certainly goes back to the dawns of celebrity (there must have been some name-droppers in philosophical circles in Ancient Greece), it’s a surprise to see it acquire its specific name so recently. And most languages probably don’t even have a good equivalent; I would certainly be hard put to give a good translation of name-dropping in French. Are there other languages better suited to the task?

I was last week in Verbania, Italy, for the conference in honor of the 150th anniversary of the Riemann Hypothesis, which was also the inaugural activity of the new RISM (Riemann International School of Mathematics). It was a very pleasant occasion. Already during the Sunday afternoon presentation (open to a general audience), there was a beautiful historical lecture by R. Narasimhan, who explained in particular that “monodromy” was invented by Riemann, in a course on hypergeometric functions which barely escaped being forgotten: only three students attended it, two of whom dropped after a few lectures, and the last one publicly stated (years later) that he hadn’t understood a word of what Riemann was saying, but had stayed because his father (or maybe someone else: I have forgotten this detail) had told him that Riemann was the new man in mathematics, and that he should follow his lectures… Which he did, taking faithful notes — though he did so in a special type of shorthand which almost made them useless a few years later when time came to transcribe them.

The main lectures were especially pleasant (for me at least) in areas of geometry which I am not usually involved with: learning the current state of the art in an interesting field of mathematics can be quite a bit more enlightening if it comes from two or three hours of lectures coming from real experts (especially if there are opportunities to discuss any question afterwards). So I particularly liked the short course by J-P. Demailly, the one by J. Cheeger, and the two lectures by C. Voisin, who explained very clearly the current knowledge of both the topological constraints for Kähler varieties (what is apparently called the “Kodaira problem”), and what is currently the best that is understood about the notorious Hodge conjecture.

The slides of many lectures can be found on a page of the conference web site, and others will be posted soon (including mine; the beamer presentation can already be downloaded here).

The conference was all the more enjoyable due to the very pleasant setting by the shore of the Lago Maggiore, and although the weather was not uniformly good, the best day was Wednesday, when the excursion to the Borromean Islands was organized. The quality of the organization can be seen from the rather fancy notebook that was given to participants at the beginning:

The Mertens formula is the name given to the asymptotic determination of the behavior of the partial products of the Euler product for ζ(s) at s=1, where the zeta function has a pole: precisely, it is the statement that

as x tends to infinity (the product being of course restricted to primes), with γ being the Euler constant:

This formula has a special place in elementary prime number theory in that, although the asymptotic behavior is not so deep (at the level roughly of Chebychev-type estimates), and it is very easy to prove, from the Prime Number Theorem, in the form

for some constant c, the fact that this constant is none other than γ is not a purely formal consequence of the Prime Number Theorem. And if one looks at the standard treatments (e.g., in Hardy & Wright, and in Tenenbaum’s book, which follow more or less the same reasoning), it’s hard to come back with a clear explanation of why this is so, and what this constant may possibly mean.

I’ve been thinking about this the last few days because of something that looked very much like a serious problem in my joint work with A. Nikeghbali (which I’ve described earlier — the paper will be posted soon) on analogies between random matrices, zeta, and the Erdös-Kác theorem. This was pointed out to us by P. Bourgade: in the post above, I said

Well, one can clearly expect a finite-field version of the mod-Poisson convergence for ω(f), where f ranges over monic polynomials of degree d tending to infinity over a fixed finite field:

where there is an innocuous constant γ_q>0, the function Φ₁ is the same as before (i.e., it comes from random permutations), and

where the product runs over irreducible monic polynomials in A (so it is completely analogous to the Euler product for Φ₂…)

and although I didn’t write down the “innocuous” constant, its value, as we had it in our paper, was given by

where the sum ranges over irreducible monic polynomials in F_q[T] and

is the number of such polynomials of degree j.

The problem with this, as Bourgade pointed out, is that if one takes u=0, then obviously everything that depends on u can be evaluated, and the purported formula becomes

and this doesn’t sound reasonable at all at first sight, in view of the expression we have for γ_q. In particular, because the origin of this constant is really the (or an) analogue of Mertens’s formula for F_q[T], namely

this sounded particularly implausible (to me at least) since it did not seem reasonable to expect that the very much simpler arithmetic of polynomials over a fixed finite field could lead so easily to the same highly transcendental-looking constant that occurs for the ordinary primes: recall that the analogue of the zeta function is the much simpler function

So we looked for mistakes elsewhere, and this being the nature of preprints, we found a few minorish ones around this part of the proof, which however didn’t lead to any resolution of that problem.

And in fact, there is no problem: it is true that γ_q=1 for all q. It is psychologically amusing that I convinced myself of this not by searching Google for “Mertens function field” (which immediately gives suitable references), but by deciding to test numerically the behavior of the expressions

for various d and q: the convergence to 1 is unmistakable.

At that point I did look for references (it being even less likely to be new than the remark on the spectrum of multiplication operators). Possibly from “post-internet bias”, the earliest reference found is a paper of M. Rosen from 1999, in the J. Ramanujan Math. Soc., which unfortunately I have not been able to access. However, at least three other papers contain a proof (but I have only looked at two of them): this one by M. Car (2007), this one by P. Lebacque, and a possibly unpublished version by K. Conrad which is quoted by M. Car. Judging from the (short) Math Review, it seems that Rosen adapts the classical proof, and M. Car certainly does the same.

However, having these references is only a double-check: since our original argument was, in fact, correct, the remark of Bourgade amounts to saying that we have a (different) proof of the Mertens formula for polynomials over a finite field, in the form above:

And because of the way this is done, we understand much better where the constant comes from. In fact, there is really no constant: the right quantity to consider for product over irreducible polynomials of degree at most d is

and for primes up to x, it is

And this factor is explained by the similar behavior of random permutations: it arises because the probability that a random permutation of n letters has no cycle of length at most d is approximately

when n gets large. (For d=1, this is the density of derangements, which is well known to be asymptotically 1/e).

In fact, this should have led us to understand immediately that our constant γ_q has to be 1, since this explanation of the way the Mertens formula looks is explained (by analogy with permutations, but without proof) in the survey of Granville which I had already mentioned in the earlier post (see the “Academic’s aside” on the bottom of page 5, and the discussion before). Our own proof is very much related: we relate polynomials with no small factors to permutations without short cycles, and use suitable equidistribution statements to get the same density.

Today’s Word of the day for the OED is Power function (go there fast, as the page will disappear after one day!).
The first quotation given is from 1914, which seems a bit late, so it wouldn’t be surprising if simple searches were to yield earlier instances (as happened for algorithms).

I have to admit I am still not quite happy with my typical workflow when I write my posts. The web-based text editor of WordPress (as it exists on the ETH blogs, at least) is not very convenient (or efficient), but on the other hand writing HTML in emacs, as I would like to do, doesn’t work so well because it is does not seem possible to preview the inline LaTeX formulas from emacs.

So I’ve looked for alternatives. And I was surprised to see that on Linux, it is not so easy to find a solution. There are a number of blog editors, which would be more or less satisfactory (including, of course, an emacs package…), except that most seem to take the point of view that one should type a blog post entirely, then publish it, and then make only minor corrections (if any). But I usually try to save locally what I type after each line (almost), and I consider it important to be able to such local copies for backup and reference purpose.

After trying a few programs, the only one I found to work reasonably (and that I use now) is the fairly obscure QTM. The text editor is not the best in the world (it has a search feature, but no replace), but one can save the posts to text files (which contain some very easily understandable and removable metadata), and publish them separately later on. One fairly obvious feature of the editor (which is unaccountably missing from the WordPress editor) is the possibility to insert a link — for which the target has been previously selected in a web browser — by a simple command (CTRL+U), after highlighting the words that should refer to the link. (On WordPress, the highlighting has the effect or replacing the link address in the clipboard with the linking words; this is standard X11 behavior, of course, but here this turns out to be a drawback).

Of course, the program itself does not preview the LaTeX formulas, but this can be done decently enough by sending the post as draft to the ETH blog. A bit more annoying is the fact that I don’t see how to insert pictures in the WordPress account from QTM itself, and then refer to them directly from its editor. So sometimes I write posts without pictures and add them in from WordPress later.

There’s also an apparent bug where sometimes the publishing date sent to WordPress gets wrong, and I have to manually change that. I haven’t tried to investigate deeply if I can locate it and correct in the source code.