150 and 36

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.