E. Kowalski’s blog

One can define the -th Legendre polynomial in many ways, one of the easiest being to use the generating function

Like many “classical” special functions (which one might call “the functions in Whittaker & Watson” — I find it charming, incidentally, that the PDF of this edition has exactly 628 pages), these can also be defined using representation theory. This is done by considering the group and its -dimensional irreducible representation on the space of homogeneous polynomials in two variables of degree , where the action of is by linear change of variable:

for

Then, up to normalizing factors, “is” the matrix coefficient of for the vector . Or, to be precise (since a matrix coefficient is a function on , which is 3-dimensional, while is a function of a single variable), we have

for the elements

(the inner product used to compute the matrix coefficient is the -invariant one on ; since this is an irreducible representation, it is unique up to a non-zero scalar; the normalizing constant involves this as well as the normalization of Legendre polynomials.) For full details, a good reference is the book of Vilenkin on special functions and representation theory, specifically, Chapter 3.)

Note also that, since is, up to a scalar, the only vector in invariant under the action of the subgroup of diagonal matrices, one can also say that is “the” spherical function for (with respect to the subgroup ).

This seems to be the most natural way of recovering the Legendre polynomials from representation theory. Just a few days ago, while continuing work on the lecture notes for my class on the topic (the class itself is finished, but I got behind in the notes, and I am now trying to catch up…), I stumbled on a different formula which doesn’t seem to be mentioned by Vilenkin. It is still related to , but now seen as a representation of the larger group (the action being given with the same linear change of variable): we have

where is some other normalizing constant, and now are unipotent elements given by

It’s not quite clear to me where this really comes from, though I suspect there is a good explanation. Searching around the web and Mathscinet did not lead, in any obvious way, to earlier sightings of this formula, but it is easy enough to get thoroughly unenlightening proof: just use the fact that

expand into binomial coefficients, use the formula

for the invariant inner-product, and obtain a somewhat unwieldy polynomial which can be recognized as a multiple of the hypergeometric polynomial

which is known to be equal to (Obviously, chances of a computational misake are non-zero; I certainly made some while trying to figure this out, and stopped computing only when I got this nice interesting result…)

In 1905, the famous rower W. Burnside (then aged 52) proved one of the results known as Burnside’s Theorem (the other one being, usually, the striking result that finite groups of order divisible by at most two primes are solvable):

Let k be an algebraically closed field, and let

be a subgroup of the invertible matrices of size n over k. Let k[G] be the span of G in the matrix algebra M(n,k) of size n. Then G acts irreducibly on kⁿ if and only if k[G]=M(n,k).

Here, recall that irreducibility (a notion apparently first introduced by Burnside himself) means that there is no proper non-zero subspace

such that G leaves W invariant (globally).

This result turns out to play a role in a current research project (with O. Dinai), and since I had never looked properly at the proof(s) before, I’ve been a bit curious about it, and tried recently to understand it. There are very simple proofs known, but the shortest ones seem to be typically not very enlightening when it comes to understand why the result is true. They’re the kind of arguments you might feel you could find once you knew the result, but why would you think of proving it first? So — it was vacation time! — I had a look at Burnside’s original paper. This can be found here; if you do not have access to the Proceedings of the L.M.S, here is a fairly representative extract of the style:

As far as I’m concerned, this is barely recognizable as meaningful mathematics, and almost unreadable. I say almost, because (vacation effect) I took it as an intellectual challenge to try to reformulate Burnside’s argument in more modern terms, and I believe that I succeeded. It was a big help that the paper is only four pages long; it turns out that the one page from which the extract is taken, although I can’t explain it in any reasonable way, contains the last step of Burnside’s argument. From the fact that he needed seventeen lines to prove the “obvious” half of his theorem, there was therefore every chance that whatever is done here in one page should not be too difficult to figure out with some thought.

So here is a sketch of my reading of his proof of the non-trivial direction (that, for an irreducible action, we have k[G]=M(n,k); for full details, see this short note). We denote

and then we define

which is the linear space of all linear relations satisfied by the matrices in the group. By linearity and duality, we see that the goal is to show that, if G is irreducible, then the space R is zero. The steps for this are:

R is a subrepresentation of E’ for a natural action of G on E’ given by

in duality-bracket notation (this is not the same as the usual tensor product of the tautological representation of G on V and its contragredient);

Hence any irreducible subrepresentation in R is obtained as the image of a G-equivariant embedding

But all such maps

are of the form

for some fixed vector v in V;

But if we then specialize the definition of α with A being the identity matrix — which is in G –, we find, from the fact that the image of α is in R, that

for all λ, and this is a contradiction with the condition that v be non-zero… Hence we must have R=0.

It is not obvious here where it is necessary to use the fact that k is algebraically closed, but this is hidden in an application of Schur’s Lemma. Interestingly, it seems that Schur published this result also in 1905. Since Burnside also uses it without any comment (or hint of proof), it must have been known (at least to him) before. It is also amusing to note that, in fact, there is no mention whatsoever of a base field in Burnside’s paper.

I like this proof, in part because it would make sense to try to proceed in this way, even if the result turned out to be different (say, a characterization of the relation module R instead of a proof that it is zero). Also, I may be influenced by the similarity with the study of relations between roots of polynomials that can also be done using elementary representation theory of the Galois group, as discussed in this old blog post.

But as I said, there is a part of Burnside’s paper I really don’t understand, even if I suspect it is equivalent or very similar to what I did. And I am forever thankful that Emmy Noether came along some years later to put algebra on a more reasonable track than endless talk of “successive sets of symbols with the same second suffix” (which sounds almost like one of those alliterative exercises used to detect drunkenness…)

Somewhat later than I had hoped, I have updated the script of my spectral theory course. The version currently found online is complete as far as the material I intended to put in is concerned, but there are a few places where I haven’t written down all details (in particular for the proof of the Weyl law for the Dirichlet Laplace operator in an open subset of Euclidean space). I am also aware of quite a few small problems in the last chapter on Quantum Mechanics, due partly to notation problems (for the Fourier transforms, and for “physical” versus mathematical normalizations). I will need to re-read the whole text carefully to correct this; on the other hand, thanks to lists of corrections that I have already received from a few students, the number of typos is much less than before… I will however continue updating the PDF file as I continue checking parts of the text.

What delayed this version for a long time was the write-up of the last section on “The interpretation of Quantum Mechanics”; of course it’s in some sense an extraneous part of the script, since spectral theory barely enters in it, but I found it important to at least try to connect the mathematical framework with the actual physics. (This partly explains all the reading I’ve done recently about these issues). It is equally obvious that I am not the most knowledgeable person for such a discussion, but after all, there are good authorities that claim that no one really understands this question anyway…

What I end up discussing contains however one little mathematical result, which is cute and interesting independently of its use in Quantum Mechanics; it is a theorem of S. Kochen and E.P. Specker which states the following:

There does not exist any map

where S² is the sphere in R³ with the property that, whenever

are pairwise orthogonal unit vectors, we have

or in other words, two of the three values are equal to 1, and the other is equal to 0.

How this result enters into discussions of the interpretation of Quantum Mechanics is described by M. Jammer in his book on the subject (not the same as his book on the development of Quantum Mechanis, but another one, equally evanescent as far as the internet is concerned); more recently, J. Conway and S. Kochen have combined it with the Einstein-Podolsky-Rosen argument (or paradox) to derive what they call the “Free Will Theorem”, which is an even stronger version of the unpredictability of properties of Spin 1 particles (those to which the Kochen-Specker argument applies). Conway has given lectures in Princeton on this result and its history and consequences, which are available as videos online.

Coming back to the result above, considered purely from the mathematical point of view, it is interesting to notice that both the original proof and the version used by Conway-Kochen (which is due to A. Peres) show that the hypothetical map does not exist even for some finite sets of points on the sphere. It is of some interest to get a smallest possible set of such points. The proof I gave in the script, however, which is taken from Jammer’s book (who attributes it to R. Friedberg) is maybe theoretically slightly more complicated, but it is also somewhat more conceptual in that one doesn’t have to be puzzled so much at the reason why one finite set of vectors or another is really fundamental.

One of the undetermined numbers in Les nombres remarquables is the Mills number (or numbers; this is not uniquely defined, as will be clear from the description below). I had somehow forgotten all about it, although I have now the memory that it was quite popular in the olden days (at least, I seem to remember that it cropped up in every other conversation back when I was reading that book 20 or more years ago), and I had not heard anything about it for about that long.

So, the Mills number is that (or any of the) amazing real number A>1 with the property that

is a prime number for every positive integer n.

As one can expect, the doubly-exponential growth means that it would be pointless to try to use this to produce prime numbers. And one may guess that the proof of the existence of such a number has little to do with primes, and should apply to many other sequences of positive integers.

This is indeed so, but not in a completely trivial manner. More precisely, what the proof shows is that, given an infinite subset S of integers, and a real number c>1, one can find B, depending of course on the set and on c, such that

provided the set S has the property that, for some real number

and all large enough x, the intersection

is not empty. In other words, since θ<1, there must be some element of the set in all “short” intervals (from some point on), where “short” has the usual meaning in analytic number theory: the length is a power less than 1 of the left-hand extremity.

(Note that the relation between c and θ shows that, if we know a suitable value of θ for S, then we can always find a value of c that works, always assuming θ<1.)

What about primes, then? Do primes exist in short intervals? The answer is, indeed, yes, and it has been known to be so since the work of Hoheisel in 1930, but this is by no means a triviality! Indeed, if one looks at the problem from too far away, analyzing the number of primes in such an interval with the “explicit formulas” in terms of zeros of the Riemann zeta function, then one gets the impression that one will prove

(which is the expected answer, because of the Prime Number Theorem) only for θ>1/2, and only by knowing that

which we know only for θ=1. This means that, from the point of view of immediate consequences of the location of zeros of the zeta function, having primes in short intervals is comparable with having a zero-free strip.

From this point of view, we see that the existence of the Mills number is quite an interesting fact. Moreover, the smaller the value of c one can take, the shorter the intervals we manage to find primes in. The value c=3 which I quoted at the beginning is possible because the current best result about primes in short intervals states that, for x large enough,

contains the “right number” of primes. (In fact, this allows any c>12/5). This result is due to Huxley, and hasn’t been improved since 1972; however, if one wants only the existence of a positive proportion among the right number of primes, Baker and Harman have the record value 0.534 (this was in 1996, and allows c>2.14…).

All the proofs since Hoheisel’s time depend crucially on a way to get around the Riemann Hypothesis known as “density theorems” for the zeros. This is a fairly inconvenient name, since “density” might suggest “lots and lots of zeros everywhere”, whereas the intent and purpose of density theorems is to show that, although there might be zeros off the critical line, or even close to 1 (which is were they would fight against the pole of the Riemann zeta function, which is the White Knight that tries to produce primes, glorious primes), there can not be too many. The precise argument is presented in Chapter 10 of my book with H. Iwaniec. Note that density theorems have many other applications: certain particularly subtle ones for Dirichlet characters (“log-free density theorems”), the first of which was proved by Linnik, are crucial to the known proofs of his marvelous theorem according to which, for some absolute constant C>0, the smallest prime P(q,a) congruent to a modulo q, for a coprime with q, satisfies

(The best result here allows you to take any C>5.5 for q large enough, due to Heath-Brown; the Generalized Riemann Hypothesis gives this for any C>2). If this remains too mundane — some people do not like primes in arithmetic progressions –, note that you need similar theorems for cusp forms to give an upper bound of the right order of magnitude for the rank of the Jacobian J₀(q) of the modular curve X₀(q) for q prime, a result of P. Michel and myself.

Now for the proof of the existence of the Mills number, in the generality of a set S containing elements in short intervals. I won’t give all details, but here’s a sketch:

(1) define b(1) to be the smallest element of S above the point after which all short intervals contain at least one element of S;

(2) define inductively b(n+1) to be such that

(3) show, using the condition

that if we define

we then have

and deduce that the limit B of x_n exists, and gives the desired general Mills number…

(The title of this post is based on WBGO‘s nice late-Sunday show “Jazz from the archive”, which I used to follow when I was at Rutgers; the physical archive in question was that of the Institute of Jazz Studies of Rutgers University; the post itself is partly motivated by seeing Gil Kalai’s examples of his own “early” problems…)

The following problem of classical analysis of functions of one-complex variable has puzzled me for a long time (between 15 and 20 years), though I never really spent a lot of time on it — it has never had anything to do with my “real” research:

Let f,g be entire functions, and assume that for all r>0 we have

What can we say about f and g? Specifically, do there exist two real numbers a and b such that

for all z?

(Actually, the “natural” conclusion to ask would be: does there exist a linear operator T, from the space of entire functions to itself, such that Tf=g and such that

for all entire functions φ; indeed, the function g=Tf satisfies the condition any such “joint isometry” for the vector space of entire functions equipped with the sup-norms on all circles centered at the origin; but I have the impression that it is known that any such operator is of the form stated above).

My guess has been that the answer is Yes, but I must say the evidence is not outstandingly strong; mostly, the analogue is known when one uses L² norms on the circles, since

if a_n are the Taylor coefficients of f. If those coincide with the same values for another entire function g (with coefficients b_n) we get by unicity of Taylor expansions that

for all n, and the linear operator

is of course a joint isometry for the L² norms on all circles, mapping f to g.

The only result I’ve seen that seems potentially helpful (though I never looked particularly hard; it was mentioned in an obituary notice of the Bulletin of the LMS, I think, that I read rather by chance around 1999) is a result of O. Blumenthal — who is today much better known for his work on Hilbert modular forms, his name surviving in Hilbert-Blumenthal abelian varieties. In this paper from 1907, Blumenthal studies the structure of the sup norms in general, and computes them explicitly for (some) polynomials of degree 2. Although the uniqueness question above is not present in his paper, one can deduce by inspection that it holds for these polynomials at least.

(I wouldn’t be surprised at all if this question had in fact been solved around that period of time, where complex functions were extensively studied in France and Germany in particular; that the few persons to whom I have mentioned it had not heard of it is not particularly surprising since this is one mathematical topic where what was the height of fashion is now become very obscure).