E. Kowalski’s blog

The Mathematical Department of ETH was saddened last week of learning of the death of Beno Eckmann, on Tuesday, Nov. 25, aged 91. He was one of the major historical figures in the department, linking us to the “classical” times of Polya, Weyl, Hopf and others by his long activity and involvement in the international mathematical community. In Zürich, part of his importance has been as the initiator and first director of the Forschungsinstitut für Mathematik in 1964.

He was a student of Heinz Hopf at ETH, obtaining his degree in 1942, and his genealogy lists 73 students and 1040 descendants (!), among whom can be recognized many well-known names from topology (e.g., M. Kervaire, 1956, or G. Mislin, 1968), analytic geometry (H. Grauert, 1956), algebra (e.g, M.A. Knus, 1967) and even probability theory (E. Bolthausen, 1973), and so on. (Part of the genealogy is presented in a more impressive way in this genealogical tree).

Although I didn’t meet him after my arrival in January, my colleagues have told me he was active until quite recently (and indeed, his last research paper on Math Reviews is dated 2004, though it is claimed to originate from a 2002 lecture..). His 90th birthday was celebrated last year at FIM.

Topologists in particular will probably enjoy some of the essays collected in these notes, in particular the reminiscences of the antesagittarian days of algebraic topology. (Apparently, the very first occurence of a “physical” arrow to indicate a map between spaces is to be found in a Research Announcement by W. Hurewicz from 1941).

My colleague Richard Pink had the idea of illustrating the matrix I had found with F. Jouve and D. Zywina, whose characteristic polynomial has Galois group W(E₈), by plotting it on a grid with colors to indicate the size of the coefficients. This representation was then constructed by Leopold Talirz using Matlab, with the following result:

The square grid represents the matrix graphically, and the scale for colors is indicated on the right. I think this is an intriguing picture. Can anyone suggest an explanation for its structure? In fact, since the matrix arises by a product

it would actually probably be even more interesting to show the evolution of the matrix from the identity as more terms are added in the product (and even to continue beyond this matrix in a random walk, as this the motivation for the construction…). If this leads to a nice result, I’ll post it later…

One of my favorite constructions in the English language is that bizarre form of comparative that makes it possible to speak of the “Shorter Oxford English Dictionary”, without any mention of what this estimable dictionary (two long and heavy volumes…) is actually compared to. Does this grammatical construction have a name? Does it exist in other languages? Certainly it is completely inexistent in French, and makes for rather thorny translation puzzles: how should a number theorist translate, in French, the name of Gallagher’s remarkably clever larger sieve? [The construction is actually particularly twisted here, since the implicit comparison point of Gallagher is, of course, already known as the large sieve...]

For those readers who have never heard of the larger sieve, here is the idea and the explanation for the name (which is very clearly explained in Gallagher’s paper): recall that a basic sieve problem (for integers) is to estimate the number of integers remaining from (say) an interval

after removing all those n which, reduced modulo some prime p in some set (for instance, all those up to z=N^δ for some δ>0) always stay away from a given subset Ω_p of primes: in other words, one wishes to know the cardinality of the sifted set

Classically (and also not so classically), the first examples were those were one tries to get S to be essentially made of primes, or twin primes, etc. In that case, the size of Ω_p is bounded as p grows. There situations are called small sieves.

Then Linnik introduced the large sieve which is efficient for situations where the size of Ω_p is not bounded, and typically grows to infinity with p: basic examples are the set of quadratic residues (or non-residues), or the set of primitive roots modulo p.

And then came the larger sieve: Gallagher’s method works better than the large sieve when Ω_p is extremely large, so that the integers in S have few possible reductions modulo primes (roughly speaking, the larger sieve is better when the number of excluded classes is larger than half of the residue classes modulo p; so quadratic non-residues are borderline, and indeed both the large and the larger sieve give the correct upper bound — up to a constant — for the number of squares up to N). More precisely, Gallagher shows that

where

and

provided the denominator D is positive.

As the number of classes excluded increases, the efficiency of this inequality becomes extremely impressive: if

with θ>0, the number of elements of S becomes at most a power of log(N), whereas the large sieve gives a power of N. For an arithmetico-geometric application of a new variant of the larger sieve in number fields in a situation where the numerology is of this type, you can read a recent paper of J. Ellenberg, C. Elsholtz, C. Hall and myself.

[I should mention that it was C. Elsholtz who first mentioned the larger sieve to me a few years ago: the method is not as well known as it should, since it is extremely simple -- Gallagher deals with it in nine lines, and our version is not much more complicated, though it is a bit more involved since it works with heights in the number field to sieve elements which are not necessarily integers. The basic argument and its applications can provide excellent exercises and problems for any introductory number-theory course.]

Speaking of divergent series, I’d like to mention one of my favorites bits of informal folklore: this starts with a slogan

The only “truly” divergent series is the harmonic series

which, of course, deserves plenty of quotes. But the idea is quite sound: it is that, given about any other series with real or complex terms (which does not involve this one in a hidden manner, of course, e.g., as an “irreducible summand”), such as

it is quite often possible, by some kind of trick or feint, to assign a “value” (i.e., a real or complex number) to the series in a way which is reasonable and useful for certain purposes. [A famous example is the (apparently) ultra-divergent series

which was considered by Wallis and Euler: its “value” is

(see this article for an explanation of some of Euler’s ideas to do this “computation”).]

One setting in which the philosophy above has been refined to a precise (conjectural) statement is the theory of L-functions (over number fields). Indeed, observe that (formally) we have

where ζ(s) is the Riemann zeta function, which is only defined properly for the real part of s larger than one by the series

Then, after translating the basic insight about the harmonic series from Dirichlet series to L-functions, one gets the following folklore conjecture:

If a reasonable (say, automorphic, or “motivic”) L-function L(s) over a number field k has a pole at s=1 (when normalized so that the functional equation relates values at s and 1-s), then L(s) is divisible by the Riemann zeta function, in the sense that

does not acquire poles at any of the complex zeros of the Riemann zeta function.

This is quite a rich problem: it contains a famous conjecture of Artin (the Dedekind zeta function of a number field should be divisible by the Riemann zeta function), and applied to the Rankin-Selberg convolution, it suggests the existence of the symmetric-square L-functions of modular forms — indeed, the first crucial result towards the “automorphic” existence of the latter (due to Gelbart and Jacquet), was the proof of this divisibility property by Shimura).

Now for the most amazing thing concerning this folklore property (at least to me): it seems that it also works modulo primes! Let me explain this (an explanation which I heard from Jean-Pierre Serre): another renowned formula of Euler for the values of the zeta function at even integers can be rephrased, after using the functional equation

as the formula

where the Bernoulli numbers B_2n are rational numbers defined by the power series expansion

(and, of course, the middle expression for ζ(1-2n) is purely formal: it is one more example of a divergent series that can be given a convincing value).

The idea now is that the last expression (-1/2n B_2n) can be reduced modulo a prime p, provided p does not divide 2n and does not divide the denominator of the Bernoulli number. Now, lo and behold, the primes dividing the denominators of Bernoulli numbers are known: they are exactly the primes such that

or equivalently such that

for all integers k coprime with p (those for which the inverse modulo p makes sense…). So the Bernoulli number can not be reduced modulo primes exactly when (either p divides 2n or) formally we have

the divergent harmonic series again! (One must omit the terms with k divisible by p of course, but since this is purely formal, why not?).

I have no idea if there is a good explanation for this coincidence, but it is remarkably beautiful, and it certainly gives a convincing argument for the fact that the numerators of Bernoulli numbers are much more mysterious: so are the zeros of the Riemann zeta function, in comparison with its pole…)

Here it is:

Yes, it’s a divergent series, but I’m sure Euler would like it even more. (Actually, the probability that this formula is not somewhere in his works, or in Ramanujan’s, is close to zero, though I came upon it fairly accidentally today — maybe I’ll explain how it came about naturally at some later time).

Amusingly, both Pari/GP (numerically, using sumalt) and Maple (symbolically, after setting _EnvFormal:=true;) can confirm the “formula” as-is… (I didn’t try with Mathematica).