Consider an open disc contained in the region (other compact regions may be considered, for instance an open rectangle). For any real number , we can look at the function on . This is a holomorphic function on , continuous on the closed disc . What kind of functions arise this way? Bagchi proved the following (this is essentially Theorem 3.4.11 in his thesis):
Theorem. Let denote the Banach space of holomorphic functions on which are continuous on the closed disc. For , define a probability measure on to be the law of the random variable , where is uniformly distributed on . Then converges in law, as , to the random holomorphic function
,
where is a sequence of independent random variables indexed by primes, all uniformly distributed on the unit circle.
This is relatively easy to motivate: if we could use the Euler product
in , then we would be led to an attempt to understand the probabilistic behavior of the sequence , viewed as a random variable on with values in the infinite product of copies of the unit circle indexed by primes. This is a compact topological group, and the easy answer (using the Weyl criterion) is simply that this sequence converges to the Haar measure on . In other words, the random sequence converges in law to a sequence of independent, uniform, random variables on the unit circle. Then it is natural to expect that should converge to the random function , which is obtained formally by replacing by its limit .
Bagchi’s proof is somewhat intricate, in comparison with this heuristic justification, especially if one notices that if is replaced by a compact region in the domain of absolute convergence, then the same idea applies, and is a completely rigorous proof (one need only observe that the assignment of an Euler product
to a sequence of complex numbers of modulus one is a continuous operation in the region of absolute convergence.)
The proof I give in my script tries to remain closer to the basic intuition, and is indeed less involved (it avoids both a use of the pointwise ergodic theorem that Bagchi required and any use of tightness or weak-compactness). It makes it easy to see exactly what arithmetic ingredients are needed, beyond the convergence in law of to the Haar measure on . Roughly speaking, it goes as follows:
Now pick some parameter , and write
,
where
Fix . For some fixed big enough, is less than by Step 3, and is at most . For this fixed , tends to as tends to infinity because of the convergence in law of to — the sum defining the truncations are finite, so there is no convergence issue. So for all large enough, we will get
proudly as French and German books, or going down
as English or American or Italian books? When books are ordered by topic or author, this leads to rather uncomfortable switches of orientation of the head as one scans bookshelves for the right oeuvre to read during a lazy afternoon.
Actually, these are more or less contemporary examples, and it seems that these conventions change with time. For instance, I have an old English paperback from 1951 where the title goes up instead of down:
Another from 1962 goes down. When did the change happen? And why? And how do other languages stack up? Is it rather a country-based preference? Are the titles of Italian-language books printed in Switzerland going up (like the French and German ones do), or down? And does this affect the direction in which shivers run along your spine when reading a scary story of murdered baronets in abandoned ruins?
(There’s of course the solution, admittedly snobbish, of writing the title and author’s name horizontally
as the Pléiade does, for instance).
]]>The email address from which it came ( jlb@limbo.ow ) is probably not genuine, so I wonder who the author could be (the final note “Translated, from the Spanish, by H.A.H” is of course suggestive, but one would then like to see the original Spanish…)
]]>The meeting is organized by the Forschungsinstitut für Mathematik, and (again as usual!) there is a certain amount of funding for local expenses made available by FIM for young researchers (graduate students and postdocs). Please register on the FIM web page before March 21 if you are interested!
]]>The negative comments embody the perfect distillation of the perverse puritanical hair-splitting competition known as “Is this question a good fit for MO?” (to be read in a slightly hysterical voice) that is now what I find most annoying on the site. This is not what mathematics (not even “research” mathematics, that seems to replace here the “pure” mathematics illusion of yesteryears) is about for me. I must confess to finding particularly annoying that some of the most vocal critics (e.g., the pseudonymous “quid”) seem to be people with little actual mathematical contributions and too much time to spend and to write for ever and ever on the finer points of etiquette of a web site as if it were some platonic object to protect from all interlopers.
What would Arnold think of this discussion, where “mathematicians” throw away much (he would say “most”) of the whole history, motivation and insights of their science? Would a question of Kolmogorov on what the brain looks like as graph have passed through the fourches caudines of Signor Quid?
]]>Roughly speaking, I concentrated on a small number of examples of statements of convergence in law of sequences of arithmetically-defined probability measures: (1) the Erdös-Kac Theorem; (2) the Bohr-Jessen-Bagchi distribution theorems for the Riemann zeta function on the right of the critical line ; (3) Selberg’s Theorem on the normal behavior of the Riemann zeta function on the critical line ; (4) my own work with W. Sawin on Kloosterman paths. My goal in each case was to do with as little arithmetic, and as much probability, as possible. The reason for this is that I wanted to get and give as clear a picture as possible of which arithmetic facts are involved in each case.
This led to some strange and maybe pedantic-looking discussions at the beginning. But overall I think this was a useful point of view. I think it went especially well in the discussion of Bagchi’s Theorem, which concerns the “functional” limit theorem for vertical translates of the Riemann zeta function in the critical strip, but to the right of the critical line. The proof I gave is, I think, simpler and better motivated than those I have seen (all sources I know follow Bagchi very closely). In the section on Selberg’s Theorem, I used the recent proof found by Radziwill and Soundararajan, which is very elegant (although I didn’t have much time to cover full details during the class).
I have started writing notes for the course, which can be found here. Note that, as usual, the notes are potentially full of mistakes! But those parts which are written are fairly complete, both from the arithmetic and probabilistic point of view.
]]>The basic strategy to get this result is not new: it was devised by Fouvry and Michel a number of years ago (inspired at least in part by earlier work of Friedlander-Iwaniec and by the Vinogradov-Karatsuba-style “shift” method to estimate certain short exponential sums). What was missing (despite the strong motivation provided by applications that were known to follow from such a result, one of which is described in a recent preprint of Blomer, Fouvry, Milicevic, Michel and myself) was a way to prove certain estimates for (complete) sums over finite fields, of the type
where
unless the parameters are in some “diagonal” positions. And we cannot afford too many diagonal cases…
The main contribution of our paper (much of which comes from the ideas of Will!) is to find a relatively robust approach to such estimates.
This relies, as one can expect, from extensive algebraic-geometric arguments to apply the Riemann Hypothesis over finite fields. In fact, from this point of view, this paper is by far the most complicated I’ve ever been involved in. We use, among other things:
Many of these are results and ideas that I was aware of but had never actually used before, and I learnt a lot by seeing how Will exploited and combined them. I will try to write a few more posts later to (attempt to) explain and motivate them (and how we use them) from an analytic nunber theorist’s viewpoint. The theory of vanishing cycles, in particular, should have many more applications in extending the range of applicability of Deligne’s Riemann Hypothesis to problems in analytic number theory.
The paper is dedicated to Henryk Iwaniec, who has been over the years the most eloquent and powerful advocate for a deeper use of the work of Deligne (and Katz and others) in applications to analytic number theory.
]]>Being an island, Ventotene is reached by boat. One of the interesting things about the trip to Ventotene was to observe how birds
would follow us until a very definite point, and then suddenly disappear.
Also, the instructions to launch a lifeboat are rather daunting…
The organizers had scheduled a lot of free time during the conference besides the scientific programme. I was therefore able to take a few pictures, such as some of the local lizards,
and some of the wonderful local cats.
The end of the week also coincided with the beginning of the ten-day long celebration of the island’s patron saint, Santa Candida. Among the festivities were the evening launches
(over a week) of huge hot-air balloons (“Mongolfieri”), with some fireworks
(note the amusing effects of my camera’s fireworks scene setting without tripod…). Some Galois-theoretic persiflage was also notable…
Update The scans of my lectures can be found here.
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