# Random matrices, L-functions and primes: the final day

And now for the last day of the Conference… We started the morning with a talk of P.O. Dehaye, who explained his method (joint with Borodin) for computing averages of values of derivatives of characteristic polynomials of random unitary matrices. This approach generalizes to this case the method of Bump and Gamburd (as explained in Bump’s talk) and is based, besides partition and representation theory, on a theorem of Okounkov and Olshanski (involving shifted Schur functions). The talk ended with a suggestion that one should look closely at the Riemann zeta function (and other L-functions) from the point of view of symmetric functions of infinitely many variables, either from the Euler product over primes, or from the Hadamard product over zeros. This is quite a striking idea; certainly, from the analytic number theory point of view, primes are not really symmetric, because in most applications they do come up ordered with the information contained in their size. But maybe this ordering should be less important for certain problems — and the supply of problems in analytic number theory is not close to depletion…

The next talk, by H. Iwaniec, had been announced as Rational and p-adic zeros of ternary quadratic forms; however, as described in the previous instalment, the fall of a coin led him to change the topic and title to Asymptotic large sieve and moments of zeta and L-functions. It was a careful account of his recent work with B. Conrey and K. Soundararajan where, for the first time, an unconditional confirmation was obtained for one of the moment conjectures for families of L-functions beyond the fourth moment. More precisely, they have succeeded in proving the following asymptotic formula:

$\sum_{q\geq 1}{f(q/Q)\int_{\mathbf{R}}{\sum_{\chi}{|L(1/2+it,\chi)|^6}g(t)dt}\sim 42abcQ^2\frac{(\log Q)^9}{9!}$

where χ runs over even primitive Dirichlet characters modulo q, and both f and g are bump functions, f being supported on [1,2] and the other analytic with rapid decay at infinity, and the constants which appear are

$a=\prod_p{(1-p^{-1})^5(1+5/p+5/p^2+14/p^3-15/p^4+5/p^5+4/p^6-4/p^7+1/p^8)},$ $b=\int{f(x)xdx},$ $c=\int_{\mathbf{R}}{g(t)|\Gamma(1/4+it/2)|^6dt}.$

In fact, quite spectacularly, this is obtained as a special case of a more general theorem involving both shifts and with complete lower-order terms (a polynomial in log q) and power-saving. Iwaniec explained very carefully the statement of the general conjecture which is confirmed by this result, and discussed briefly the main new technical innovation: an asymptotic large sieve formula for certain specific coefficients, which breaks the limit imposed by the usual large sieve inequality for Dirichlet characters.

Iwaniec mentioned that the 8th-power moment analogue is barely outside of their reach, and it might also be accessible with more work. He also very briefly explained the need for the (aesthetically displeasing) smoothing in t which is involved in the statement (and should be unnecessary): it is used to obtain cancellation in expressions of the type

$\left(\frac{n}{m}\right)^{it},$

unless n and m are of comparable size.

After this impressive result, the main talks of the conference concluded with a lecture by H. Montgomery who reviewed very nicely various instances of subtle combinatorics involved in performing computations of moments of arithmetic quantities of interest. One of the basic examples is found in the derivation of the expected Poisson-distribution for the number of primes in intervals of the type

$[n,n+c\log n]$

which Gallagher derived conditionally, based on a uniform Hardy-Littlewood conjecture for prime k-tuples (for some more on this, my paper on averages of singular series may be useful). Montgomery also discussed his work with Soundararajan on understanding the deviations between the naive probabilistic model for the number of primes in larger intervals and what is expected based on various conjectures (such as the Pair Correlation Conjecture for zeros of the Riemann zeta function).