# Random matrices, L-functions and primes: the conference, live

The conference “Random Matrices, L-functions and primes” which A. Nikeghbali and myself are co-organizing with the support of the Forschungsinstitut für Mathematik started yesterday, and the second day just ended with a colloquium lecture by N. Katz about “Simple things we don’t know”. Before explaining (hopefully, tomorrow) what were some of these simple things he mentioned, here are a few words about yesterday’s lectures (the schedule is visible online).

Since a fair number of talks were given using various forms of beamer presentations, we have started gathering the corresponding files to make them available. There are currently three talks on the web page (from yesterday’s lectures):

(1) my own introductory lecture, which was intended to present the basic techniques and problems of analytic number theory to those members of the audience whose expertise lies in the direction of probability theory;

(2) Ashkan’s own not-so-introductory lecture, which is devoted to limit theorems in probability theory from the point of view of our recent joint work with J. Jacod about what we call mod-Gaussian convergence;

(3) J. Keating’s discussion of some classical and some new aspects of the connection between Random Matrix Theory and the Riemann zeta function. His emphasis was in large part on the issue of predicting full asymptotic formulas, with lower order terms, for many of the quantities for which Random Matrix Theory had first given predictions restricted to “main term” behavior (e.g., the density functions for the pair correlation of zeros, or the asymptotic of moments of the zeta function on the critical line). The point of these investigations (which lead to some quite complicated formulas which may be discouraging to look at!) is that the resulting conjectures become much more testable numerically; this is very clear in the pictures that Keating showed (based on joint works with a number of people, including prominently E. Bogomolny, B. Conrey, D. Farmer, M. Rubinstein and N. Snaith, in various combinations). The point is that, for instance, the graph of a polynomial

$p(x)=0.0001 x^2+0.1 x-12$

does not look even remotely like that of the main term

$q(x)=0.0001 x^2$

unless x is very large, and hence hypothetical experiments based on this simple main term would be very misleading for those values of x.

Keating’s discussion was continued partly in B. Conrey’s talk (the slides for which will be available soon), who explained briefly some of the methods used for these recent precise predictions (based frequently on the so-called “ratios” conjecture). He also mentioned a number of problems suggested by these results, one of which is the following particularly striking conjecture of M. Watkins: consider, among positive integers m, those which are not divisible by the cube of any prime and are congruent to 1 modulo 9, and let S be the set of those values which are sums of two rational cubes:

$x^3+y^3=m$

(for instance, m=1729). Fix a prime p>3, say p=7, and two integers x and y modulo p (say x=2, y=3). Then Watkins conjectures that the limit

$\lim_{T\rightarrow +\infty}\frac{|\{m\in S\,\mid\, m\equiv x\text{ mod } p,\ m\leq T\}|}{|\{m\in S\,\mid\, m\equiv y\text{ mod } p,\ m\leq T\}|}$

exists, and predicts its value; in particular, for the example of p=7, x=2, y=3, the limit should be 2. In other words, among cubefree integers, those which are congruent to 37 modulo 63 should be about twice as likely to be a sum of two cubes as those which are congruent to 10 modulo 63!

Conrey said he found this particularly beautiful, and it is hard to disagree! The statement is completely elementary, despite being a prediction that is completely dependent on quite deep mathematics, involving elliptic curves, their L-functions and the Birch and Swinnerton-Dyer Conjecture in particular, and ideas of Random Matrix Theory. The papers of Conrey, Keating, Rubinstein and Snaith and of Watkins on quadratic and cubic twists of L-functions explain how all this comes about.