I’ve mentioned earlier the first cases I had found of (almost) mod-Cauchy convergence (see this post for the definition).

Yesterday, following from a citation in Sarnak’s note to one in a paper of Guivarc’h and Le Jan, I ended up looking at this paper of F. Spitzer, which implicitly proves a true mod-Cauchy convergence result. Morever, this result is very interesting from the probabilistic point of view, since it concerns that most beautiful of objects, the (complex) Brownian motion.

Here is the result: consider such a complex Brownian motion

for non-negative t, where the real and imaginary parts are themselves independent standard Brownian motions on the real line, except that the real part is started at

for some fixed R> 0 (for instance R=1).

Because almost surely the Brownian motion is not zero, one can define a continuous process

which measures the argument of the Brownian motion and its winding around the origin. Spitzer’s result is then the fact that

converges in law, as t goes to infinity, to a Cauchy distribution with parameter 1 (it had already been noticed by P. Lévy that the argument of the Brownian motion is not square-integrable).

However, his argument is more precise: using relations with suitable PDE’s, he manages to compute exactly the Fourier transform of the law of the argument in terms of Bessel functions: for u≥0 (this characteristic function is even), we have

where Iν(x) is the I-Bessel function of index ν.

Since the first term in the power series expansion (at 0) of the Bessel function is given by

it follows easily that we have a mod-Cauchy convergence:

as t goes to infinity, with parameters and limiting function given by

I’m continuing preparing my notes for the course on exponential sums over finite fields, and after the fourth moment of Kloosterman sums, I’ve just typed the final example of “explicit” computation, that of the Salié sums, which for the prime field Z/pZ are given by

(the (x/p) being the Legendre symbol, and e(z)=exp(2iπ z) as usual in analytic number theory). These look suspiciously close to Kloosterman sums, of course, but a surprising fact is that they turn out to be rather simpler!

More precisely, we have the following formula if a and b are coprime with p (this goes back, I think to Salié, though I haven’t checked precisely):

the inner sum running over the solutions of the quadratic equation in Z/pZ. The first term is simply a quadratic Gauss sum, which is well-known to be of modulus

and since the inner sum contains either 0 or 2 terms only, depending on the quadratic character of a modulo p, we get trivially the analogue

of the Weil bound for Kloosterman sums.

The standard proof of this formula is due to P. Sarnak. It is the one which is reproduced in my earlier course on analytic number theory and in my book with H. Iwaniec. However, since it involves a somewhat clever trick, I tried a bit to find a more motivated argument (motivated does not mean motivic, though one can certainly do it this way…).

I wasn’t quite successful, but still found a different proof (which, of course, is very possibly not original; I wouldn’t be surprised, say, if Sarnak had found it before the shorter one in his book). The argument uses a similar trick of seeing the sum as value at 1 of a function which is expanded using some discrete Fourier transform, but maybe the function is less clever: it is roughly

instead of something like

(and it uses multiplicative characters in the expansion, instead of additive characters). It’s a bit longer, also less elementary, because one needs to use the beautiful Hasse-Davenport product formula: denoting

the Gauss sums associated with multiplicative characters, we have

which is the analogue for Gauss sums of the duplication formula

for the gamma function. Since this formula is most quickly proved using Jacobi sums (the analogues of the Beta function…), which I had also included in my explicit computations of exponential sums, using this argument is a nice way to make the text feel nicely interlinked and connected. And it’s always a good feeling to use a proof which is not just the same as what can be found already in three of four places (at least when you don’t know those places; for all I know, this may have been published twenty times already).

Now, you may wonder what Salié sums are good for. To my mind, their time of glory was when Iwaniec used them to prove the first non-trivial upper bound for Fourier coefficients of half-integral weight modular forms (this is the application Sarnak included in his book), which then turns out to lead quite easily (through some additional work of W. Duke) to results about the representations of integers by ternary quadratic forms. Another corollary of Iwaniec’s bound, through the Waldspurger formula and Shimura’s correspondance, was a strong subconvexity bound for twisted L-functions of the type

where f is a fixed holomorphic form and χ is a real Dirichlet character, the main parameter being the conductor of the latter.

The point of Iwaniec’s argument was that the Weil-type bound, when applied to the Fourier coefficients of Poincaré series, which can be expressed as a series of Salié sums

(with fixed m and n) in the half-integral weight case, just misses giving a result. So one must exploit cancellation in the sum over those Salié sums, i.e., as functions of the modulus c. This is hopeless, at the moment, for Kloosterman sums, but the semi-explicit expression for the Salié sums in terms of roots of quadratic congruences turns out to be sufficient to squeeze out some saving…

(Nowadays, there is a wealth of techniques to directly prove subconvexity bounds for the twisted L-functions — e.g., in this paper of Blomer, Harcos and Michel –, and one can run the argument backwards, getting better estimates for Fourier coefficients from those; as is well-known, one finds this way that the “optimal” bound for the Fourier coefficients is equivalent with a form of the Lindelöf Hypothesis for the special values…)

One of my favorite computations is that of the fourth moment of Kloosterman sums:

where

This was almost first performed with spectacular consequences by H.D. Kloosterman himself in 1927, as a crucial step to proving an upper bound for his sums, which was sufficiently good for his application to representations of integers by integral positive definite quadratic forms in four variables (I recommend reading at least the introduction to this paper: it is strikingly modern).

I say almost, because after just checking his paper, I realized he just got the right order of magnitude, and not the exact formula, for M4.

The standard reference I had been using (including for the exam of a graduate course I taught a while ago…) was in Iwaniec’s delightful book on classical modular forms. (Kloosterman sums appear there because, as in fact already noticed by Poincaré in 1912, they occur in the formulas for Fourier coefficients of Poincaré series…) But while typing the result for my lecture notes of my new course on sums over finite fields, I worked out a different argument than the one in Iwaniec’s book (different but, it turns out, rather closer to Kloosterman’s own…).

Roughly, one first quickly reduces (using orthogonality of additive characters) to computing the number of solutions of the equations

This can be computed quite directly, as Iwaniec does, but one can also observe that there are obvious solutions

and wonder what others can exist? It is then fairly natural to try to see whether knowing

is enough to recover the pair (x1,x2), up to permutation. This will be the case, if we can compute the value of the product x1 x2 from the two symmetric quantities above (this is the theory of symmetric functions, in a rather trivial case). Now, observe the identity

which gives what we want, provided (of course) the denominator is non-zero. And indeed, this may of course vanish, and does so precisely for the extra solutions

of the original equations… The argument therefore proves there are no other than these three families, and after figuring out their intersections, the formula for the fourth moment follows. (Details are in my ongoing notes already mentioned above…)

This computation may seem desperately low-brow; however, as I discuss briefly in Section 6 of my most recent survey on applications of the Riemann Hypothesis over finite fields (I tend to like writing about this, I must confess…), this can be interpreted, via the “Larsen alternative” as the crucial step in proving the vertical (or average) Sato-Tate Law for Kloosterman sums: if we write

then the collection of angles

becomes equidistributed with respect to the Sato-Tate measure

as p goes to infinity…

[Update (27.2.2010): thanks to Ke Gong for sending some useful typographical corrections to the notes.]

This semester, I am teaching (besides a course on Integration and Measure theory, about which I’ll write later) a course on elementary methods in the study of exponential sums over finite fields. The intent is to describe first the proof of the Riemann Hypothesis of A. Weil for one-variable exponential sums, based on Stepanov’s method (possibly in the version of Bombieri, possibly not), then go to more recent results where the “elementary” methods put to shame the cohomological formalism, e.g. Heilbronn sums or Mordell-type exponential sums involving polynomials with large degree (as in the work of Bourgain, though I haven’t yet quite settled on the detailed programme for that part of the course).

I’m hoping to type my lecture notes as I go along. In fact, the goal of the course is partly to prepare things both for a follow-up in the next semester on the cohomological approach and for a book I’ve been thinking about for quite a while on this topic. I don’t know what will come of this idea (for one thing, I’m starting slowly and as elementarily as I can, which is not really the style of the final book I have in mind, which would be a user guide for already fairly experienced analytic number theorists and other mathematicians interested in applying exponential sum methods to their own problems), and I doubt that even two semesters will be enough to lecture on what I wish to include, but the notes will be available on this page (together with links to various other documents of interest).

There might be some readers who are currently desperately looking for a suitable epigraph for their mathematical masterwork. The best advice I can give is to spend some time in the company of Thomas Pynchon’s works, which abound in scientific and mathematical wit. Many, though aware that P.G. Wodehouse’s wonderfully more readable oeuvre is unfortunately sadly lacking for this purpose, will still object by pointing out the reputation for incomprehensibility of, say, “Gravity’s Rainbow”, a heavy volume supposedly barely more understandable than “Finnegans wake”. However, it should be kept in mind that this reputation is the work of literary critics, who — and they are more to be pitied than castigated — are unlikely to find that the veil lifts when, around page 670, the dashing Yashmeen Halfcourt of “Against the day” starts conversing cogently in Göttingen with David Hilbert to propose what is commonly referred to as the Polya-Hilbert idea to solve the Riemann Hypothesis. But this, of course, is exactly where a mathematician will think that, after all, it’s not so bad.

Here are some of my favorite quotable excerpts from Pynchon:

• From “Gravity’s Rainbow”, which is also full of Poisson processes, if I remember right:
  

  “The Romans,” Roger and the Reverend Dr. Paul de la Nuit were drunk together one night, or the vicar was, “the ancient Roman priests laid a sieve in the road, and then waited to see which stalks of grass would come up through the holes.”

  (actually, I have to confess, with respect to this citation, to having committed two of the cardinal sins of epigraphists: I’ve used it twice — my excuse being that one time was for my PhD thesis, which was not published as-is –, and I haven’t read the book much further than beyond the place where it appears; and for those who wonder, there is at least one more dreadful faux pas in epigraphing: doctoring a quote to make it just perfect — and I’ve done it at least once).

• In “Mason & Dixon”, we find

  In the partial light, the immense log Structure seems to tower toward the clouds until no more can be seen.

  This novel was published in 1997; one cannot feel anything but impressed to see Pynchon following so closely the latest developments of post-Grothendieck algebraic geometry…

• Still in “Mason & Dixon” (which I am currently re-reading, hoping to vault triumphantly above the 50 percent mark of understanding), we have

  He sets his Lips as for a conventional, or Toroidal, Smoke-Ring, but out instead comes a Ring like a Length of Ribbon clos’d in a Circle, with a single Twist in it, possessing thereby but one Side and one Edge….

  which prompts the obvious question: is it really possible to blow a smoke ring in the shape of a Möbius band? Hopefully some experts will comment on this…

Here is a quick update to the last post about (restricted) mod-Cauchy convergence; I’ve investigated numerically the behavior of the renormalized averages

(see the post for the notation) for some values of t, to see if the limitation to

in Vardi’s result could be a mere artifact of the method. Here are some graphs representing these empirical averages for

(click to see the full pictures):

In particular, note that in the last picture, the vertical scale runs from -300 to 300, more or less, compared with oscillations between 0.5 and 1.5 in the third. So it seems pretty convincing evidence that the limit as N goes to infinity does not exist when t is large.

(Note: the empirical average for N=5000 involves about 8,000,000 Dedekind sums).

I’ve already mentioned what my co-authors (J. Jacod, A. Nikeghbali) and myself call “mod-Gaussian convergence”, and “mod-Poisson convergence” (and I will hopefully soon write down some summary of the more recent developments of these notions; for the moment I’ll just mention that the first two papers will appear soon, one in Forum Mathematicum and one in International Math. Research Notices). It was obvious how to extend these definitions to other natural families of standard probability distributions, the only condition required being that the characteristic functions must not have zeros (since we need to divide by them). Precisely, consider any set Λ of probability measures on the real line, with

for any

Then we can say that a sequence Xn of random variables converges mod-Lambda for some parameters

and some limiting function Φ, if we have limits

for all real t, which we assume to be locally uniform to avoid degeneracies.

There are “easy” examples like

where Yn has law λn and is independent of the fixed random variable Y: in that case the limiting function is simply the characteristic function of Y. But one may wonder about more interesting examples, and part of the works mentioned above was dedicated in finding such examples in number theory when Λ is either the set of Gaussians or the set of Poisson variables.

Now I’d like to mention two cute examples involving Cauchy distributions; they are interesting for a couple of reasons: (i) again, they involve arithmetic, but of quite a different sort (and one could in fact be interpreted as a geometric or topologic phenomenon); (ii) they are not quite of the type above: indeed, for these two examples, the convergence only holds for t in an open interval around t=0, not for the whole real line (or rather, it’s not entirely clear whether they extend; the proofs don’t, and seem to do so for good reasons, but I haven’t yet looked very deeply at this).

I’ve written a short note with the details (short because, like some of the early cases of mod-Gaussian and mod-Poisson convergence, the results are mostly re-interpretations of earlier works; those, however, are by no means trivial), which can be found here. I’ll describe briefly one of them here, which is related to a result of Vardi on the distribution of Dedekind sums (the second is maybe even more fun: it is related to linking numbers of modular knots, a topic much popularized by É. Ghys, and based on a recent preprint of Sarnak; but I’m less familiar with the underlying objects).

First, here is the definition of the Dedekind sums, defined for coprime positive integers c and d with d< c:

where ((x)) is the saw-tooth function, periodic of period 1 with

These look strange or arbitrary when presented so bluntly, but they are quite important and fairly-ubiquitous in certain areas of mathematics (see for instance here for a report on a recent workshop dedicated to them…)

The question solved by Vardi concerned the distribution of these sums. Precisely, he proved that — suitably normalized –, the sums s(d,c), averaged over all c<N and all allowed values of d, have a limiting Cauchy distribution. To state this formally, let first

and write

and

to get some (finite) probability spaces. Then Vardi proves

for any a<b, where μ is a standard Cauchy distribution, namely

The characteristic function of the more general Cauchy distribution with parameter γ>0, namely

is given by

and since those functions do not vanish, one may wonder about the possibility of getting here some example of mod-Cauchy convergence. And indeed, if one looks at Vardi’s proof with such an ulterior motive, one sees that this is derived elementarily from an asymptotic formula for the characteristic function of s(d,c) on FN. Namely, after cleaning up the notation, Vardi proves that

uniformly for

where

and the limiting function is the rather remarkable expression given by

in which the function η(z) is the Dedekind eta-function!

This is a restricted mod-Cauchy convergence: because of the size of the parameters, the “main term” is in fact not always dominant, and the limit of

is only guaranteed to be Φ(t) in the range

Since the error term in Vardi’s formula depends crucially on the spectrum of some Laplace-like operator acting on modular forms with a multiplier system depending on t, it is however not clear at all that this can be improved to obtain a larger range. (But this may be tested experimentally, since Dedekind sums can be computed very quickly using the reciprocity relation they satisfy).

A puzzling feature of Vardi’s argument is that, although one is not altogether surprised to see the eta function coming up in studies of Dedekind sums (indeed, Dedekind sums were defined from their occurence in the transformation properties of the eta function!), the precise connection leading to this asymptotic formula is quite indirect.

Compared with our examples of mod-Poisson and mod-Gaussian convergence, one very different feature is the shape of the limiting function. At the moment, I do not know anything really about the behavior of Φ(t), in particular, the behavior of the second expression involving the integral of powers of the eta function. I haven’t found anything about them yet in the literature, but it wouldn’t be surprising if some special identities, for instance, were known…

From the CERN English website:

CERN’s Visits Service organises tours of its experimental areas and facilities, which are free of charge. Tours in several languages are organised on Mondays to Saturdays starting at 9 a.m. and 2 p.m. It is essential to book in advance.
Please note that the tours are not suitable for children under 14 years of age.

(emphasis mine, as people say in history books).

Now from the French version:

Le service des visites du CERN organise des visites gratuites de sites d’expériences et d’infrastructures. Ces visites sont proposées en plusieurs langues, du lundi au samedi, à 9h ou à 14h. La réservation est obligatoire.
Veuillez noter que le niveau des visites n’est pas adapté aux enfants de moins de 10 ans.

I’m still thinking aloud (or the bloggerly equivalent thereof) about the topic of my last post, and I’m at this delightful stage of guessing there may well be interesting questions there, and yet not knowing too precisely which ones are easy, which are impossible, or even which are already hidden in the maze of MathSciNet under cleverly disguised search terms.

So consider the case of G=SL(2,Z) again, and assume given a subgroup H. In broadest terms, we’re trying to identify which conjugacy classes in G have representatives in H. We can’t exclude that all of them do; if that happens, we know that (1) H is of infinite index (see the first comment by D. Speyer to the earlier post); (2) but H surjects, by reduction modulo p, to SL(2,Fp) for every p. The latter condition implies in particular that H be Zariski-dense in SL(2) (otherwise, its reduction would be in G(Fp) for some proper algebraic subgroup, and this would be strictly contained in SL(2,Fp) if p is large enough). Nicely enough, such subgroups (especially when finitely generated) are currently the topic of much work in terms of spectral theory, expansion and the like (see for instance these recent preprints by Bourgain, Gamburd and Sarnak, and by Bourgain and Kantorovich).

The conjugacy classes of G have been classified for a long time (for instance, this is needed for the Selberg Trace Formula). The most interesting, or at least those I’m going to look at first, are the so-called hyperbolic ones, which are characterized by the fact that, for some (unique) a>1, they contain a representative which is conjugate in SL(2,R) to

which acts as a dilation

on the Poincaré upper half-plane. A more direct characterization, in terms of an arbitrary representative g of the conjugacy class, is that

So, for instance, we can take the conjugacy class of

In the case of a conjugacy class in G, the dilation a is a real quadratic integer (it is the largest eigenvalue of the matrix, and the determinant, which gives the constant term of the minimal polynomial, is 1). In the example above, we get

In SL(2,R), the dilation is the unique invariant of a hyperbolic conjugacy class (and visibly any a>1 occurs as a dilation). In G, things get a bit more arithmetic (which means more complicated, though the two words are maybe not quite synonyms). Essentially (I am here forgetting or glossing over some important semi-technical issues), for a given discriminant

there are only finitely many G-conjugacy classes, and the number of them is the class number of the associated real quadratic field. (Precise details are given in this old paper of Sarnak).

From my point of view of conjugacy classes, the following seems the obvious salient features:

(1) to have a chance to find a given hyperbolic conjugacy class in a subgroup H, a necessary condition is that H contains a matrix with a certain trace (up to sign; if we assume that minus the identity is in H, the sign ambiguity disappears); this condition, in turn, is obviously susceptible to local congruence obstructions — but we know that for a Zariski-dense (finitely generated) subgroup of G, all but finitely many of these congruence obstructions modulo primes will vanish by Strong Approximation.

(2) if we have a subgroup where all local obstructions disappear (for instance, all reductions modulo primes are surjective; not I don’t actually have an example of a proper subgroup of infinite index where this holds…), we are led to wonder whether all ideal classes associated with hyperbolic elements of G have representatives in H; this question is reminiscent of the representation problem for integers by ternary definite quadratic forms (where there are fairly simple necessary conditions for this to happen, and those are fairly classically also sufficient for an integer to be representation by some form in the same genus as the given one, which means by some form everywhere locally equivalent to it, while the representability by the given form holds for sufficiently large integers by much deeper work involving Fourier coefficients of half-integral modular forms — a very beautiful story, where crucial work is due to Iwaniec and Duke and Schulze-Pillot).

As before, hopefully more to come…

A fairly well-known fact about finite groups says that if H is a subgroup of G, and H intersects every conjugacy class in G, then in fact H=G. This is quite useful, for instance, for some problems of Galois theory, because one might have to understand a finite group using information only about which conjugacy classes it represents in a bigger group (e.g., a Galois group represented as permutation groups of the roots of an integral polynomial, where the factorization of the polynomial modulo various primes indicates which conjugacy classes of the corresponding symmetric group intersect the Galois group; I’ve already mentioned this type of things here and here).

It is natural to ask what happens with other kinds of groups. The example of compact Lie groups shows that if G is infinite, there may well exist a subgroup H intersecting every conjugacy class; for instance, if G=U(n), every element can be diagonalized, i.e., every element is conjugate to one in the subgroup H of diagonal matrices (which, if n is not 1, is not the same as G…) However, these are quite special groups, and one might suspect that some interesting infinite groups retain this property (which I’ll call the Jordan property here, as suggested by Serre’s nice paper about this theorem of Jordan).

Although I’ve started looking around, I haven’t found much information yet on this. The first groups I’m trying to understand are arithmetic groups like G=SL(n,Z). Here’s one simple example in such a case: if n is at least 3, then G has the Jordan property “with respect to finite index subgroups” (i.e., any finite index subgroup intersecting all conjugacy classes of G is equal to G). This requires a fairly big hammer, but is otherwise very easy: by the Congruence Subgroup Property, any H of finite index satisfies

for some integer d, where

is a principal congruence subgroup. This means that, for some subgroup Γ of the finite quotient G/G(d), we have

but then it is immediate (by lifting to H) that if H intersects all conjugacy classes of G, then also Γ intersects every conjugacy class in G/G(d), and we get

from the finite group case, and therefore H=G.

More generally, one sees at least (without using the Congruence Subgroup Property) that if H is a subgroup of G=SL(n,Z) intersecting every conjugacy class, then we have

for all d (because the reduction modulo d maps G surjectively to SL(n,Z/dZ) for all d). However, this condition is not as stringent as it may look: it is known (the “Strong Approximation Theorem” of Mathews, Vaserstein and Weisfeiler) that this holds, at least for all integers d coprime with some “conductor” f, for any subgroup of SL(n,Z) which is Zariski-dense in SL(n), and such groups can be quite “small”. However, one might intuitively hope that, being “smaller” than finite index subgroups, they would intersect fewer conjugacy classes (?). On the other hand, I also don’t know offhand of a non-trivial subgroup with conductor f=1…

For the special case n=2, the Congruence Subgroup Property fails (one way to see it, as explained in this survey of Raghunathan, is to contrast the fact that SL(2,Z) has finite quotients like the alternating group A5, whereas any non-abelian simple quotient of a congruence subgroup is of the type SL(2,Z/pZ) for a prime p, and none of these is isomorphic to A5, simply because none is of order 60). Then it’s not clear to me if some finite index subgroup (not of congruence type) could intersect every conjugacy class of SL(2,Z).

Hopefully, I’ll have the occasion to write more about this as I explore the literature…

Nature, stretching Horace Davenport out, had forgotten to stretch him sideways, and one could have pictured Euclid, had they met, nudging a friend and saying: “Don’t look now, but this chap coming along illustrates exactly what I was telling you about a straight line having length without breadth.”

(taken from the first pages of Uncle Fred in the Springtime; due to a rather unfortunate fall in the stairs last week, I have to rest and watch my back for a few days, and hence I’ve been in need of light and refreshing literature to pass the time, turning therefore in part to re-reading some novels of P.G. Wodehouse.)

Every time I have found a mistake in one of my books (or paper), I have had to repeat two or three times the well-known mantra The only way to not make mistakes in print is to not publish anything, before I could regain some composure. If the mistake was indicated by a reader, it is even worse. Of course, the problem often turns out to be mostly innocuous — e.g., the definition of “equidistribution” in my book with Henryk Iwaniec is wrong (we forgot to restrict the open sets for which convergence is claimed to those where the boundary has limiting measure zero), but it’s hard to imagine much harm coming from this. However, more serious errors are more disturbing; in particular, it makes you wonder what the frequent casual claim that The responsibility for all mistakes of course lies with [the author] really means in practice, when honest readers may well be led to a lot of misguided effort because of what is claimed with characteristic authorial confidence.

To give an unfortunate example, B. Conrey just pointed out that the same book has a very unfortunate claim that the local factor of the Rankin-Selberg convolution of two classical (holomorphic or Maass) modular forms is “the obvious one” for a ramified prime p which appears with exponent exactly one in the l.c.m. of the conductors of the two factors. In particular, this means that we claim that if f (resp. g) has p-factor

corresponding (in the case of elliptic curves) to split/non-split multiplicative reduction at p, then the local factor for the Rankin-Selberg convolution is

which is in particular of degree 1 in p-s. Alas, alas, this is quite wrong; the local factor should be of degree 2! The simplest way to see this is probably to think in terms of the local Langlands correspondence (note that one doesn’t need to know it is a theorem to apply it heuristically): the local factors for each form are supposed to be of the type

where ρ is a 2-dimensional representation of some local Galois group acting on V, Fp is the Frobenius at p, and I is the inertia group. Generically one might indeed assume that if ρ1 and ρ2 have each a one-dimensional invariant subspace, the tensor product (which corresponds to the Rankin-Selberg convolution) would also have the same property (with basis

of the invariant subspace, where e corresponds to that of ρ1 and f to that of ρ2. But the classification of these things (which are among the so-called Steinberg representations) shows that it is possible (it might indeed be always the case: I need to brush up my understanding of all this before making claims here!) that

is a twist of ρ1 by a character of degree 1. Then this means that one can find a basis e, f of a common underlying two-dimensional space so that I acts by

and then, of course, we see that both vectors

in the tensor product are invariant under I.

As I said, this mistake is quite annoying. My guess is that it may not have created any trouble (yet) for our readers: I’m pretty sure that the claim we make is true if the prime p is ramified only for one of the two modular forms (I’ll have to find a proper reference, of course), and I don’t think many analytic applications would have been outside this case. However, I plan to look at least quickly through the list of papers on MathsciNet which refer to our book to detect possibly problematic cases…

As pointed out by Roman Holowinsky, a Google search for “Sarnak” gives very amusing results… Apparently, this is the name of a character race on some video game (which I had never heard about); see (for instance) here for more, including pictures and many hilarious quotes…

Some recent and upcoming publications, which will be a good opportunity to exercise the unnumbered list HTML tag…

• The journal Mathematika, which is very well-known at least among analytic number theorists, and which had almost no internet presence (and a rather haphazard publication schedule) until very recently, is now distributed online and on paper by Cambridge, which also promises that its archives will become available soon. Since this is where many of the foundational papers and applications of the large sieve — including Bombieri’s groundbreaking paper where he proves his version of the Bombieri-Vinogradov theorem — were published (due to the influence of Davenport, certainly), I am particularly happy…
• The long-awaited book of Friedlander and Iwaniec on sieve methods is announced by the A.M.S, for a May publication date.
• Also upcoming from the A.M.S, in March, is a book of essays and surveys of Poincaré’s work in Mathematics and his influence. This is an English translation of a French book. It contains in particular a (short) chapter on Poincaré et la théorie analytique des nombres that I wrote. The English translation is not mine; in fact, my initial reaction on receiving the English text (for checking and proofreading) was a very strong dislike, and even (almost) rejection! Somehow, the fact that I hadn’t written those words, and yet I was supposed to be the author of that chapter, had a very surprising psychological effect on me. This reaction passed, though only after I produced my own translation for my own satisfaction, and — of course — realized it was much of a muchness compared with the other one. But I can now understand much better the extreme problems that may occur in literary translation, and I sympathize with the ambivalent feelings that may arise then among authors. I also wonder if my friend W. Appel reacted equally deeply to my own English translation of his mathematics textbook for physicists…

I’ve changed the theme of the blog; the main reason was that I had become annoyed enough at how the previous one insisted in displaying italics in bold to look for an alternative. The new one’s default fonts are (maybe) a bit too small, but it is easy enough to make them a bit bigger in most browsers.

As expected at the end of the earlier post, I have found my copy of the detailed programme of the performance of “MSI: The anatomy of integers and permutations”, and I can therefore give the names of the actors involved.

First, sitting on chairs on the stage, from left to right in the picture,

we have:

* Lorraine Wochna, as The Narrator;

* Emily Ann Barth, as Emmy Germain;

* Matthew Boston, as Professor K.F. Gauss;

* Mike Mihm (replacing Jay Stratton at the last minute), as Sergei Langer;

* Carl Wallnau, as Detective Jack Newman (or von Neumann; the title page of the programme uses the latter spelling, but the inside description spells it Newman);

Sitting on the stairs in front of the stage is

* Matthew Archambault, as Barry Bell;

Standing behind the readers, on the stage, are Jessica Manley, Michael Spencer and (not visible) Jennifer Granville.

Also absent are two important characters with silent role: Joe Ten Dieck and Count Nicholas Bourbaki.

Last Saturday evening, in Wolfensohn Hall at the Institute for Advanced Study, was held the first semi-staged reading of the mathematical screenplay “MSI: Anatomy”, written by Jennifer Granville and her brother Andrew Granville (with original music by R. Schneider and stage design by M. Spencer).

I won’t even try here to summarize or describe the story (but I will say that it was a great success), being no media critic and hardly an expert on mathematical detective fiction (a proper review by V. Miller will appear, I am told, in the Notices of the AMS), but I am happy to mention that most of the mathematics discussed can also be found in Granville’s very entertaining survey on analogies between integers and permutations. I am also told that efforts to present the story to a wider audience are in progress…

Here is the poster:

And here is a picture taken by C.J. Mozzochi at the beginning of the performance; the blackboards, besides pictures of cheese, include excerpts of excellent mathematics, some of which faithffully reproduced from B. Green’s lecture (given two days before) on his work with T. Tao and T. Ziegler on the inverse conjecture for Gowers norms:

(I apologize for not including the name of the fine actors involved: I have misplaced my copy of the programme, where they are listed, but I will update the post once it is found again — such a simple act of detection should be within my feeble means…)

(For those who missed them, the first day was about π, and the the second day was about ζ(2); what will the third day reveal… read on!).

According to Hadamard’s factorization for the “completed” zeta function

(which is an entire function of s, and is invariant under the replacement of s by 1-s), we can write

where the product runs over all non-trivial zeros of the Riemann zeta function (those in the critical strip), and a and b are some constants.

Now, as in Euler’s original definition of the Gamma function, it is tempting to replace the exponential terms by

which leads to

with

(All these products converge absolutely for all s since the series with general term |ρ|2 converges). Now, since

,

(easily remembered because the zeta function has a simple pole with residue 1 at s=1), we can plug in the value s=0 and s=1 to get

which leads to what may be anachronistically called Euler’s second product for the Riemann zeta function:

(I have never seen this formula before, but of course it is very unlikely to be new!)

One may even mix this with Euler’s gamma formula

and the well-known value Γ(1/2)=π1/2, which together give the nice expression

from which we can incorporate the trivial zeros of the zeta function at -2, -4, -6, etc, in the product, and deduce

where the product runs now over trivial and non-trivial zeros!

Now, it is more than tempting to specialize the value of s; the first formula, for s=2, leads to

or in other words to the relation

where the product is — once more — over non-trivial zeros. Again, I would bet this has already appeared somewhere, but I had never seen it.

Using the table of the first 10000 positive ordinates of zeros of zeta (found here), one gets (putting in the complex conjugates, of course) the values

with the first 10, 100, 1000 and 10000 zeros, respectively, compared with π/3=1.04720.

I think I’ve found the mysterious author of the notes on 3-manifolds: it is (or should be) G. P. Scott. The crucial clue is the fact that the notes claim that the author, and Shalen independently, proved that “3-manifold groups are coherent”, and then gives the proof. This would immediately clarify things, were it not for the fact that (1) Shalen never published his proof; (2) the terminology “coherent” doesn’t seem to be really well known for groups, really. What it is defined to mean is the following: a group G is coherent if and only if, all its finitely generated subgroups are finitely presented.

But, as it happens, even Scott’s paper proving this doesn’t seem to use the terminology! (In MathSciNet, there are ten papers by someone named Scott including “coherent” somewhere in the review — but again that one is not among them) Fortunately, Google did find some references for “Shalen coherent”, in particular a Bourbaki seminar by J. Stallings reporting on Scott’s result (which gives, in particular, simple examples of non-coherent groups).

[Note: On Scott's page, I found what seems to be a quite nice survey of The geometries of 3-manifolds, from 1983.]

As long as written texts remain an important part of mathematics, we can expect that — every once in a while — boxes or bins will appear in a common room, or in a library, or outside some retiring professor’s office, with an enticing “Please take” or Servez-vous to encourage the random walker (or flâneur, or Spaziergänger) to pick up some old preprint or other. Thanks to such open-ended generosity, my own collection has been enriched by an old textbook I’ve already discussed, a fair number of Bourbaki Seminar reprints, and a few mimeographed reprints from André Weil’s own collection (also, a somewhat melancholy sight, an italian translation of his sister’s play Venise sauvée, or “Venice saved”), including lecture notes of Siegel, de Rham and papers of Serre and Ihara, with a few (unfortunately rather benign) marginal notes.

Monday last week, as I was at the University of Pennsylvania (to give a lecture in their Algebra and Galois Theory seminar — video accessible from Ted Chinburg’s web page…), I found a few such inviting bins in the common room. I quickly picked up what seems to be a very nice set of lecture notes (or survey?) of 3-manifold topology, dating apparently from the mid-seventies. In particular, it being typewritten (or xeroxed from a typewritten original), I grabbed it with especial promptness, thinking that this might well be a text that is not really available anywhere else.

However, I can’t quite confirm this because there is no indication of the author’s name, either at the beginning or at the end of the set of notes. Googling the first sentence (”The basic problem of manifold theory is that of classification”) didn’t bring any hit. But maybe some readers will recognize it? Here’s a picture of the first page, for all 3-detectives…