E. Kowalski’s blog

One of my mathematics teachers, a long time ago, once objected to statements of the type

Let X and Y be two compact topological spaces. Then X x Y is also compact

on the ground that the use of two implied that the statement did not apply to the case of X x X, whose compacity would need to be stated separately, as it was not, strictly speaking, an application of the given statement.

His favored solution was to drop the two (or, in French, to replace Soient X et Y deux espaces… by Soient X et Y des espaces….), with the idea (I presume) that making a grammatical mistake (using a plural form like des when, sometimes, there is only one object, if X=Y) would be less important than a mathematical one.

Strangely enough, I still sometimes remember this, and I have modified various sentences to try to go around it, although the whole thing seems quite absurd really… I wonder if others have heard this type of rules, and if there’s a mathematically and syntaxically correct way to phrase things without being absurdly formal?

“Every sum is a trace” is a well-known folklore saying in certain automorphic circles (echoed by a no less convincing “Every sum is an expectation” around probabilists); in this spirit, let’s have a look at one of the most famous sums

which Euler was the first to evaluate.

It is possible to see it, and then compute it, as a trace, and I’m sure this has been done many times; here is (a variant of) the way I presented things for an exercise for my Spectral Theory class.

Consider the Hilbert space

(for the Lebesgue measure), and the Volterra operator T:

which is a linear operator acting on H, in fact a Hilbert-Schmidt operator with kernel

which is bounded, and therefore certainly belongs to the space

The operator T is therefore compact, and the operator S=T^*T is also compact, and in fact positive, so that the trace is well-defined as a non-negative real number, or infinity. The trace is well-known to be expressible in different ways: (i) as a sum of the series formed with the eigenvalues of S (with multiplicity); (ii) as the sum of the series

for an arbitrary choice of orthonormal basis (f_n)_n of H; (iii) as the integral

This last integral is of course completely elementary: it is the area of the lower-half triangle in the square [0,1]² below the diagonal; in other words, it is 1/2.

For an alternate expression (hence an identity), we look at the series above for the easiest orthonormal basis available:

For the special case n=0, we have

hence

For non-zero n, we have

and therefore (Parseval, if you wish, or direct computation):

Summing over all n and identifying the two expression for the trace, we get

and hence — unsurprisingly, I presume — we get

(I said unsurprisingly, but I first managed to get confused enough about the computation — for a slightly different operator — that, for a while, I almost convinced myself that ζ(2)=π²/12).

As a proof (which, I repeat, is certainly not new, though it is not found in this collection), this is fairly close in flavor to the Fourier-expansion proofs, where one expands (typically) the function x-1/2 on [0,1] into Fourier series before applying the Parseval identity. (In fact, it seems this is “dual” in some simple way which could be made precise).

Like the Fourier-expansion argument, it has the nice feature of showing almost immediately that it will be possible to generalize the argument to

for k> 0 integer, using T^k instead of T; it also hints quite strongly that the result will be of the type

for some rational number α_k. But it is equally obvious that this will not work at all like this for zeta evaluated at odd positive integers, as it should…

A select few of my mathematical books exhibit the type of quirky behavior that (quite justifiably) causes authors to consider publishers as being in league with the devil. In increasing order of amusement, here is one page of the index of my copy of Reed and Simons’s “Functional Analysis” (Modern Methods of Mathematical Physics, Vol. I)

Reed and Simon index

Then here is one page of Goodman and Wallach’s “Representations and invariants of the classical groups”

Page of Goodman-Wallach

which almost looks normal, except for (as in the red circle I drew) the ligatures “fi” which are missing. It must have caused much grinding of teeth to the authors to note that this is not the case all over the book: many of the pages contain an abundance of “finite”, “definition”, etc, with no error whatsoever. In particular, opening the book at random, you would never detect the problem.

And finally, my masterpiece, if I may say so: my copy of Katz and Sarnak’s “Random matrices, Frobenius eigenvalues, and monodromy”, where the introduction, from page 5 to page 20, felt that its importance justified that it be repeated after page 228 (up to page 244):

Two pages of Katz-Sarnak

None of these, however, are as extraordinary as the instance reported in the story “The Missing Line” of Isaac Bashevis Singer, where an abstruse philosophical sentence – “the transcendental unity of the apperception” — mystically moves from one Yiddish newspaper to another. (Although it is in a work of fiction, so might be a complete invention, I have the impression that it is so bizarre that it must have actually happened).

A few remarks about searching:

(1) Like many (probably most) researchers, I use search tools on the internet multiple times daily to locate relevant information to what I do; these searches, however, are only as good as the search terms I use, and every once in a while, it is really hard to guess the right ones! The last instance of this is that it is only last week that J. Achter pointed out to me (and my coauthors) an interesting paper of D. Masser from 1996 which is very closely related to my paper on simple jacobians in families and the larger sieve (with J. Ellenberg, C. Hall, and C. Elsholtz; see also Quomodocumque’s post on this).

(2) Apart from Google, of course, the most useful tool for a mathematician is the Mathematical Reviews (and Zentralblatt, though I have to admit that I rarely look at the latter). I can still remember when this only existed in paper, and I would sometimes browse the Number Theory section when the new monthly issue came in; and I remember even more vividly the first electronic version (around 1993–94), on six (!) CDROMS, which required a multi-platter CD-ROM reader/changer to be accessed. There was one workstation with it in the library of the Institut Fourier, and the search times were atrocious because of the delays in changing one disc for another… Now my question is: does there exist a similar database in other fields? Specifically, I’ve been searching high and low for prior references to hand signals in geckos (the power of blogging: this is the third hit when searching google for “gecko”…), and I’ve found that being deprived of the analogue of Math Reviews makes me feel quite helpless. I haven’t found any explicit reference on Google Scholar, and I don’t know if there is somewhere a better source of information on such a topic.

(I did search for “Gecko” in Math Reviews: no luck… however, if you’ve discovered a nice mathematical object which seems very sticky and acrobatic, you know a good name for it now…)