We will discuss recent results concerning the problem of establishing rigorous moment estimates and subconvex bounds for L-functions of large degree.

#
50 Years of Number Theory and Random Matrix Theory Conference

Sixth and higher moments of L-functions are important and challenging problems in analytic number theory. In this talk, I will discuss my recent joint works with Xiannan Li, Kaisa Matom\"aki and Maksym Radziw\il\l on an asymptotic formula of the...

In 2018 Keating, Rodgers, Roditty-Gershon and Rudnick established relationships of the mean-square of sums of the divisor function $d_k(f)$ over short intervals and over arithmetic progressions for the function field $\mathbb{F}_q[T]$ to certain...

### The recipe for moments of L-functions and characteristic polynomials of random matrices

In 2005, Conrey, Farmer, Keating, Rubinstein, and Snaith formulated a 'recipe' that leads to precise conjectures for the asymptotic behavior of integral moments of various families of $L$-functions. They also proved exact formulas for moments of...

We introduce a new zero-detecting method which is sensitive to the vertical distribution of zeros of the zeta function. This allows us to show that there are few ‘half-isolated’ zeros. If we assume that the zeros of the zeta function are restricted...

I will talk about recent work towards a conjecture of Gonek regarding negative shifted moments of the Riemann zeta function. I will explain how to obtain asymptotic formulas when the shift in the Riemann zeta function is big enough, and how we can...

Montgomery's pair correlation conjecture ushered a new paradigm into the theory of the Riemann zeta function, that of the occurrence of Random Matrix Theory statistics, as developed in part by Dyson, into the theory. A parallel development was the...

I will give an introduction to Gaussian multiplicative chaos and some of its applications, e.g. in Liouville theory. Connections to random matrix theory and number theory will also be briefly discussed.

Multiplicative chaos is the general name for a family of probabilistic objects, which can be thought of as the random measures obtained by taking the exponential of correlated Gaussian random variables. Multiplicative chaos turns out to be closely...

### Large deviation estimates for Selberg’s central limit theorem, applications, and numerics

Selberg’s celebrated central limit theorem shows that the logarithm of the zeta function at a typical point on the critical line behaves like a complex, centered Gaussian random variable with variance $\log\log T$. This talk will present recent...