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...
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...
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...
