E. Kowalski’s blog

Alas, literary glory has eluded me. In fact, it seems that most voters did not consider “the large sieve” particularly odd at all, and my book ends last in the running (or last but one, it’s not clear from the announcement).

(And if you’re wondering: yes, I am undaunted, and I do have another book in plan despite this deadly blow).

As I was preparing the lecture I just gave for the SMO Day, I came back to a book I had loved when I was a kid but which I had not spent much time reading since, François le Lionnais’s Les nombres remarquables, written with J. Brette (this is a title which is surprisingly hard to translate; “Remarkable numbers” doesn’t seem right, because it misses the subtle highlighting Les).

In any case, this is a kind of dictionary of interesting numbers, which was published in 1983. I know there exist a few others, but this is the only one I’ve looked at, so I can’t compare it with them. However, this is definitely a very nice book. Partly this may be due to the fact that le Lionnais (should one say, the author, or the collector?) was not really a professional mathematician (despite what Wikipedia seems to claim in the English article), but a type of renaissance man interested both in sciences in general and a humanist; he was the co-founder (with R. Queneau, the man behind Zazie dans le métro) of the famous Oulipo. However, there is no cute amateurishness in his selection of numbers; of course, he doesn’t disdain curiosities like

843 973 902. The largest number obtained, in the decimal system, as a product of two numbers constructed with all digits from 1 to 9. 843 973 902=9642 x 87531.

but what he (or they) think are the “most interesting numbres” are highlighted with one to three stars, and the next item shows that they know that mathematics is not all cuteness:

898 128 000. *** 2⁷ 3⁶ 5³ 7 11. Order of the McLaughlin group, the tenth sporadic group.

There are even a few complex numbers, and some “non-determined finite numbers”. Among the first, I note the following property of

which I didn’t know (or had forgotten): the triples

are the only triples of (distinct) complex numbers with modulus one such that 0 does not belong to the convex hull of

for any integer n.

Among the non-determined finite numbers, there is the one defined as follows, which I suspect is probably determined now:

where k is the digit following the first sequence of seven consecutive sevens in the decimal expansion of π. (I think sufficiently many digits have been computed now to determine N).

Today I was invited to give a short lecture on the occasion of the award ceremony for the Swiss Mathematical Olympiads (the national olympiads from which the Swiss participants to the International Mathematical Olympiad are selected). So I talked (of course…) about prime numbers. I’ve put the presentation on the web (it is in French, but of course there is only a minimal amount of text so it should be fairly understandable; if there’s any interest, I’ll prepare an English translation).

Suppose you are asked to describe an infinite sequence of signs + or -. How likely is it that you will end up with the same sequence as the one I am now thinking about? Well, according to all natural probabilistic models, the probability should be zero. Read on, then…

Quite a while ago, I wrote a short note to try to understand the distribution of the root number (also known as the sign of the functional equation) for the Jacobian of the modular curve X₀(p), where p is prime. My guess was that this should be +1 or -1 more or less equally often, and it should have been easy to confirm it by computing explicitly this sign using the Eichler-Shimura relation for the L-function of this jacobian and the trace formula for the Fricke involution, except that the latter turns out to involve class numbers, and after watching the dust settle, one sees that the approximate equidistribution is equivalent to asking some questions about the equidistribution modulo 4 or 8 of some class numbers of imaginary quadratic fields for which one knows the residue modulo 2 or 4, respectively.

And in particular, one needs to know the distribution modulo 4 of h(-p) for p congruent to 3 modulo 4, in which case genus theory says that the class number is odd.

And now for the coincidence: having reached this question, it seemed fairly natural to drop by the office (two doors up on the other side of the corridor) of one of the foremost expert on the distribution of class numbers, Henri Cohen, and ask him if the answer was already known? As it turned out, Henri was just in the midst of using Pari/GP to compute the values

of the p-adic Gamma function at 1/2. These, although they should (in theory) be elements of p-adic fields, satisfy

or, in other words, these values form a sequence of signs. And the two sequences are the same!: we have

After this, I can not be impressed when hearing of people who just happen to think of their great-aunt who has been lost in the jungle of New Guinea for twenty years just a few minutes before receiving a phone call from her.

At the FIM Tea, this afternoon, the observation of the almost-never-erased blackboards in front of the seminar room led to the following historical question of clear importance: when was the blackboard, as a tool for lectures, invented? For instance, did Newton (who was a professor at Cambridge around 1670) ever lecture on a blackboard?