Travel and vacation have delayed the writing of the next installment of my series of posts on “Algebraic twists of modular forms”, but I am working on it and hope to have it ready soon… In the meantime, here are two updates on some of my favorite topics from yesterposts.
First, for the bad news: if your proof of the Riemann Hypothesis depended on my earlier claim that the spectral gap for the Lubotzky group modulo primes is at least , then you’re in trouble. This bound depended on an induction which turns out to be so mistaken that I have promised myself never more to be upset when I see a completely wrong induction proof in a student’s paper. This was found by the referee of my paper on explicit growth and expansion, who also did the most amazing job in checking all the myriad details inherent in a paper of this type. I have put up a corrected version on the web, where the spectral gap becomes again or so. (I have also started re-reading and updating my notes on expander graphs, correcting the same mistakes, but also replacing the fully explicit version with a more readable “just show the spectral gap exists”-writeup.)
Now, for the better news: in the last few weeks, I also prepared a written version of the mini-course on “Sieve in discrete groups” that I gave at MSRI in February, on the occasion of the “Hot Topics” Workshop held there concerning “Thin groups and super-strong-approximation” (a Proceedings volume is in preparation; another excellent survey already available is the one by Rapinchuk on Strong Approximation). This was just intended to be a straightforward survey, but while finishing it, I wondered again about a question I had vaguely thought about earlier without conclusion: “Is there an Erdös-Kac Theorem in the context of the ‘affine’ sieve?” More precisely, as in the Bourgain-Gamburd-Sarnak context, let be a finitely generated subgroup, and assume (for simplicity) that its Zariski closure is (which means that, for all primes large enough, the reduction modulo from surjects to ). Let be a non-constant, integral-valued, polynomial function on . Can one prove that the number of prime factors of , for a “random” element , is approximately gaussian when suitably normalized?
The answer, it turns out, is “Yes”. In fact, as soon as I thought about it a bit seriously for a few minutes, I remembered a very nice paper of Granville and Soundararajan which gives a short and easy proof of the classical Erdös-Kac Theorem, and goes on to explain how their method can be generalized to study the number of prime factors in much more general sequences than the integers. It was then almost immediately clear that one can use this method to get a form of Erdös-Kac Theorem for discrete groups (and I wouldn’t be surprised if this had been noticed earlier by others.) One interesting point is that this seems to be a case where defining “random” elements of seems most natural in terms of random walks, instead of looking at balls with respect to some metric or other. In the situation above, if is a generating set of , we denote by a random walk on , starting at with steps taken uniformly and independently at random from . Then one shows that there exists some such that the random variables
converge to the standard gaussian as , where is the number of primes dividing , with the convention that it is for .
This applies in other contexts involving sieve in discrete groups. For instance, if is a sequence of random Dunfield-Thurston 3-manifolds, and if we denote by the number of primes such that , with the convention that if has non-zero first rational Betti number, then the sequence
also converges in law to the standard gaussian.
Of course, I can’t help wondering if there could exist, as in the classical case, a finer statement of mod-Poisson convergence concerning the limit behavior of
for . This seems very hard (taking gives the average of the Liouville function) and rather mysterious… The renormalized Erdös-Kac Theorem is really about the distribution of “small” prime factors (on logarithmic scale) of integers, as one can see easily by noting that an integer has a bounded number of prime factors for any fixed ; since the theorem implies in particular that most integers have about prime factors, we see that the limiting distribution arises from integers without prime factors of such size. The mod-Poisson convergence, on the other hand, does take these factors into account, but we have currently no idea whatsoever about the distribution of “large” prime divisors of in the affine sieve context…