E. Kowalski’s blog

The 150th Birthday of Darwin’s “On the origin of species” is being amply celebrated all over the world, but let’s not forget another scientific milestone sharing the same birthday: the Riemann Hypothesis was formulated 150 years ago, in Riemann’s famous short paper Ueber die Anzahl der Primzahlen unter einer gegebenen Grosse

in the Monatsberichte der Berliner Akademie, November 1859. This should also be celebrated in various places, one of which will be a conference held in Verbania, Italy, later this month, which I am very much looking forward to.

I will give a lecture during the conference on the topic of Some aspects and applications of the Riemann Hypothesis over finite fields. It’s during the last day, so I must admit I haven’t yet gone very far in selecting exactly which topics I will mention.

However, this led me to have another look at Deligne’s paper giving the first proof of the Riemann Hypothesis in the form conjectured by Weil. And this brings me to the second topic of this post: by a nice coincidence, Deligne’s paper

is essentially contemporary with Szemerédi’s paper

proving the existence of arbitrarily long arithmetic progressions in sets of integers with positive density. (The publication years are different, but both were submitted in 1973, Szemerédi’s on July 1, and Deligne’s on September 20, and Deligne states in the introduction that he had lectured on the proof in July in Cambridge).

To my mind, it’s hard to imagine a more striking illustration of the varieties of the arithmetic experience than these two papers. At the time, and probably still today, the number of readers able to understand both of them in depth must have been dangerously close to zero (as far as I’m concerned, I’ve looked at Deligne’s quite closely, but I still have no serious experience with Szemerédi’s theorem, alas). Both are well-known to have been the source of inspiration for many people, and new proofs of their main results have been found, and have also had enormous importance.

The two papers can also be taken as good experimental tests (at least in number theory) of the “Two worlds of mathematics” issue: ask anyone which one he or she prefers (or better, which one he or she would rather have written…), and from the answer you can probably guess pretty accurately whether the person in question considers him(her)self to be of the theory-building or problem-solving kind…

I may as well state that I am not, myself, quite sure if this type of distinction really makes sense. After all, Deligne ends his paper with two applications which can be stated completely concretely: there is the Ramanujan-Petersson conjecture, which for the Ramanujan delta function is just the statement that

for the coefficients at prime indices of the (formal) power series

and there is his estimate for exponential sums of the type

if the polynomial P in k variables of degree d has the property that the zero set of the homogeneous part of degree d defines a smooth hypersurface (take

and n>1 if you want to try your hand at a specific case…)

It is probably another interesting poll, from the sociological point of view, to ask: “Which do you think is deepest?” (Partly because so few answers will come from people who can really judge, so there will be a divide between those who answer, say: “Deligne’s”, because they understand it better, and the other is just combinatorics, after all; and those that say: “Szemerédi’s”, because they understand Deligne’s work better, and therefore the mysterious incredible combinatorial games and the theorem of Szemerédi must be mystically deeper).

This question of depth, interestingly, was probably considered “solved” at the time the papers appeared: compare the (well-deserved) long and detailed review of Deligne’s paper by N. Katz (“This is without question the most important paper in algebraic geometry to have appeared in the last ten years… Deligne has proved the Riemann hypothesis for varieties over finite fields!”), with the seven lines devoted to Szemerédi’s paper (still laudatory, of course: “By an exceedingly ingenious and complicated elementary method the author proves the following theorem, thus settling a celebrated conjecture…”).

Here are two further remarks on these papers: (1) both authors acknowledge that they wrote the papers using auditor’s notes (Katz, in the case of Deligne, and Graham and Hajnal, in the case of Szemerédi’s); and both have quite short reference lists (15 references for Szemerédi, and 8 for Deligne).

Finally, I hadn’t realized that Szemerédi’s paper was published in the volume of Acta Arithmetica in memory of Linnik. Since, like many analytic number theorists, I believe that Linnik is one of the greatest mathematicians of the 20th century, but is somewhat under-appreciated (compared with his achievements!), I found this a well-deserved tribute.

Here is a fundamentally obvious observation which must be well-known, but which seems to be overlooked in the three or four texts of functional analysis I’ve looked at (for instance, here). This came up during my Spectral Theory class). Consider a (bounded) multiplication operator M_g acting on

where (X,μ) is a finite measure space, i.e., we have

where the multiplier g is a measurable bounded function. A basic question is to understand the spectrum of M_g, which is the set

where “invertible” is meant in the Banach algebra of bounded/continuous linear maps from H to itself (which means the same as “bijective” here because M_g-λ is continuous and H is a Hilbert space, so by the Closed Graph Theorem for instance, its set-theoretic inverse, if it exists, is continuous).

The intuitive answer is that the spectrum should be the image of g in C; however, this can’t be right in general, because the spectrum must be closed, and there is no reason that g(X) be closed. So the standard correct answer is that

the essential range (or image) of g, defined by

This is a fine answer as it goes, but there is a feeling that one should minimize the number of definitions involving too many epsilons and quantifiers, if possible. And the observation is that this is perfectly possible here: this definition exactly means that

is the support of the image measure; we recall that this measure is defined by

for any measurable set A in C, and that a point λ belongs to its support if and only if any of its open neighborhoods, say the discs with radius ε>0 around it, have (strictly) positive measure, which exactly translates to

namely, this is equivalent to λ belonging to the spectrum of M_g, as claimed.

This coincidence means in particular that there is no need to try to remember how to prove that this essential range is closed in C: everyone knows that the support of something has to be closed…

Another point where this is useful is in the definition of the functional calculus for these multiplication operators: suppose

is given and is continuous. The (continuous) functional calculus defines a bounded operator f(M_g) in a natural way, which means that if f is a polynomial, this operator is the “obvious” one. Since one checks immediately by induction and linearity that

if f is a polynomial, it is natural to expect the same formula to hold in general. The problem with this, is that it is not necessarily the case that

since g is only measurable (and bounded), so the composition is not immediately well-defined. But the point is of course that

(using the characterization of the support as the complement of the largest open set with zero measure), so the composition makes sense “almost everywhere”.

In this interpretation, the spectral mapping theorem

becomes the fact that

(which makes sense because f is continuous and the support is compact here, so its image under f is closed, as it should be); this can be checked directly, of course.

Note: The reason why multiplication operators are important is, of course, that the spectral theorem shows that any normal operator on a Hilbert space is unitarily equivalent to an operator of this type.

As most readers have probably learnt now, Tim Gowers’s “Tricki” is now accessible on the internet. In particular, my earlier article on Smoothing sums can be found there now, and consequently the older page I had will not be updated anymore with new material or corrections (but I have added a link to the other version). A look at the tricki’s markup language will explain that it’s much nicer to write and keep in shape these articles on the tricki than it was in the mixture of HTML and LaTeX I was using.

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…

And now for some algebra, for a change… One of my teachers (not the pedantic one, in fact a fairly well known topologist) once told us that Homological Algebra should be learnt outside of any class, either alone in one’s room, or with friends (this is in keeping with Lang’s sole exercise in the corresponding chapter of some editions of his Algebra, which asks the reader to just take any book on the topic and prove every statement without looking at the proofs).

If you’ve followed this type of advice, there is a fair chance that you’ve only ever proved the Snake Lemma and its friends for categories of modules over a ring. As it turns out, this is not so restrictive, since some abstract theorems show that any abelian category can be embedded in such a simple one, but it might be argued that this is not very elegant, especially if the theorems in question are taken for granted without proof (or reference).

All this to say that T. Bühler gave a lecture today at our Algebra and Topology seminar, where he explained some of his recent paper (to appear in Expositiones Math.) giving complete detailed proofs of all standard diagrams and diagram chasing lemmas, starting from scratch (or more precisely from axioms for exact categories, which seem quite a bit more general than abelian categories). As he remarked at the end, doing it this way is actually shorter, and it is much more satisfactory.