Number Theory

We discuss the local statistics of zeros of $L$-functions attached to Artin--Scheier curves over finite fields, that is, curves defined by equations of the form $y^p-y=f(x)$, where $f$ is a rational function with coefficients in $F_q$ ($q$ a power of~$p$).
We consider three families of Artin--Schreier $L$-functions: the ordinary, polynomial (the $p$-rank $0$ stratum) and odd-polynomial families.
We present recent results on the $1$-level zero-density of the first and third families and the $2$-level density of the second family, for test functions with Fourier transform supported in suitable intervals. In each case we obtain agreement with a unitary or symplectic random matrix model.

I will talk about recent work joint with Nathan Ng and Peng-Jie Wong. We established an asymptotic formula for the eighth moment of the Riemann zeta function, assuming the Riemann hypothesis and a quaternary additive divisor conjecture.

We compute a first moment of $GL(3)\times GL(2)$ $L$-functions twisted by a $GL(2)$ Hecke eigenvalue at a prime. We talk about the ideas behind the proof, ways in which it can be generalised or extended, and obstacles for doing so in other directions. We also talk a bit about why such moments are interesting, briefly discussing some applications.

Let $f$ and $g$ be holomorphic cusp forms for the modular group $SL_2(\mathbb Z)$ of weight $k_1$ and $k_2$ with
Fourier coefficients $\lambda_f(n)$ and $\lambda_g(n)$, respectively. For real $\alpha\neq0$ and $0<\beta\leq1$, consider a smooth resonance sum $S_X(f,g;\alpha,\beta)$ of $\lambda_f(n)\lambda_g(n)$ against $e(\alpha n^\beta)$ over $X\leq n\leq2X$. Double square moments of $S_X(f,g;\alpha,\beta)$ over both $f$ and $g$ are nontrivially bounded when their weights $k_1$ and $k_2$ tend to infinity together. By allowing both $f$ and $g$ to move, these double moments are indeed square moments associated with automorphic forms for $GL(4)$. These bounds reveal insights into the size and oscillation of the resonance sums and their potential resonance for $GL(4)$ forms when $k_1$ and $k_2$ are large.

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 Radziwi\l\l~on an asymptotic formula of the sixth and the eighth moment of Dirichlet $L$-functions averaged over primitive characters mod~$q$ over all moduli $q \leq Q$ (and with a short average over critical line for the eighth moment). Unlike the previous works, we do not need to include an average on the critical line for the sixth moment, and we can obtain the eighth moment result without the Generalized Riemann Hypothesis.