For anyone who has an interest in sieve methods, I would like to point out that my new book


is now available from Cambridge University Press (and fairly extensive preview is found on Google books)…  In fact, I’ve just received my first copy.

Of course, even before it appeared (but too late to incorporate the changes), I had found a few typos and mistakes, so a list of corrections is necessary and available; it will be updated as needed.

If you’re not quite sure if you are interested in sieve, I suggest a short look at this guest post on T. Tao’s blog (as well as Tao’s own discussion of the parity problem, though the latter is concerned with the so-called “small” sieves, which have a distinct flavor compared with the “large” sieve which I consider in the book).

Four letter words

Here’s another funny question about the English language: what is the sequence of three letters XYZ (where X, Y and Z are among the 26 letters of the alphabet) such that the number of correct (English) words of the form TXYZ is maximal, where T is just another letter? In this, it seems also best to assume that the last letter Z is not “s”, since the latter allows for many plurals of what are really three-letter words.

I think that, depending on the dictionary used (in particular, archaic words can be quite common among short words like this, and will only appear in the OED), the answer is between 13 and 17. The dictionary on my laptop (aspell-0.60-en) gives one word with 13 (say TWR), two with 12 (say BGA and IPX), and a few ending with “s”. The ending TWR jumps to 17 when permitted to appeal to the OED, and so does BGA. However, IPX only goes to 13.

Answers next week if nobody tries his/her hand first at the solution…

Mathematical (science)-fiction

Suppose there existed a natural probability (or density, or weakening of such) on the “set” of all models of the Zermelo-Fraenkel set-theory axioms. Suppose some “natural” mathematical statement had the property of having positive probability, different from 1, of holding in a random model. How should we interpret such a situation? Say, if the Continuum Hypothesis has probability 6/π2, of being false?

And if some natural statement P was shown to be a consequence of two other statements, having probability p and q, respectively, of holding in a random model, with p+q>1… so that the existence of a model where P holds would follow in highly non-constructive fashion… What would you think, philosophically or intuitively, of the “truth” of that statement?


What does it say of the psychology of English-speaking people that, according to the Oxford English Dictionary, they can say sympathetic in (at least) three additional languages without leaving the confines of theirs? Indeed, we read:

(1) sympathisch, a.

Also erron. sympatisch. [Ger.: see SYMPATHIC a.] =SYMPATHIQUE a.

(2) sympathique, a

[Fr.: see SYMPATHIC a.] Of a thing, place, etc.: agreeable, to one’s taste, suitable. Of a person: likeable, en rapport with one, congenial. Cf. SYMPATHETIC a. 2b.

(3) simpatico, a.

Also (fem.) simpatica. [It. or Sp.: see SYMPATHIC a.] Pleasing, likeable; congenial, understanding; sensitive, sympathetic.

(My impression was that “simpatico” is Italian rather than Spanish, but another dictionary gives “Simpático” for the Spanish translation of “sympatisch” and for the Portuguese translation, so if the accent can be omitted, this makes five languages for the price of three…)

There are of course copious supporting quotations; the best is

“There is something simpatico about Pascal; he is a kind of Central European Baron Munchausen.” (A. Huxley, 1969).

though this one is close:

“I do think, when you get to my age, dear, there is something sympathique about a wig, don’t you?” (E. Waugh, Vile Bodies).

Questions for all friends of alphabets, syllabaries and other dictionaries: Are there examples, in English or another language, of words with more translations allowed? In fact, are there any more translations of sympathetic in the O.E.D?

One shouldn’t disturb binomial identities too much

When teaching the most elementary courses at the university, a little bit of combinatorics enters, and the relation between the binomial coefficients and the expansion of (x+y)k, for non-negative integers k. An often appreciated trick for the students is to prove some identities among binomial coefficients using “analytic” properties of polynomials, and interpret them combinatorially, or conversely. The most basic of these identities is probably


In the spirit of fun, assume we think of rewriting this as


And now, maybe while whistling idly to pretend that we are not doing anything, let’s jolt the denominators with a quick flick of the finger:


Since it seems that no one has noticed anything, let’s do it again:


and again, and again… but stop! a red-faced policeman comes, and says that he has nothing against some good clean fun, especially on Boat Race Night, but enough is enough, and what horror have we done with poor Newt’s lovely identity:


Well, of course, what is behind this is an undoubtedly well-known identity, which can be expressed in terms of hypergeometric functions with very general parameters. But the sliding denominators might be a nice thing to show students, and the (or at least, one elementary one) actual proof of the actual formula is a good exercise in exploiting “polynomiality” in various forms, so it can be used for that purpose…