# Scientific

## The eighth moment of the Riemann zeta function

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.

## Geodesic restrictions of Maass forms and moments of Hecke $L$-functions

How large are the $L^2$-restrictions of automorphic forms to closed geodesics? I will discuss how this problem can be shown to be equivalent to proving bounds for certain weighted moments of Hecke $L$-functions, and how the lattice structure of the ring of integers of real quadratic numbers fields can be exploited to obtain essentially optimal upper bounds for these weighted moments.

## Twisted first moment of $GL(3)\times GL(2)$ $L$-function

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.

## Moments and periods for $GL(3)$

The celebrated Motohashi phenomenon concerns the duality between the fourth moment of the Riemann zeta function and the cubic moment of automorphic $L$-functions of $GL(2)$. Apart from its structural elegance, such a duality plays a very important role in various moment problems. In this talk, we will discuss the generalized Motohashi phenomena for the group $GL(3)$ through the lenses of period integrals and the method of unfolding. As a consequence, the Kuznetsov and the Voronoi formulae are not needed in our argument.

## Double square moments and bounds for resonance sums for cusp forms

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.

## $L^p$-norm bounds for automorphic forms

A fundamental problem in analysis is understanding the distribution of mass of Laplacian eigenfunctions via bounds for their $L^p$ norms in terms of the size of their Laplacian eigenvalue. Number theorists are interested in the Laplacian eigenfunctions on the modular surface that are additionally joint eigenfunctions of every Hecke operator---namely the Hecke--Maass cusp forms. In this talk, I will describe joint work with Peter Humphries in which we prove new bounds for $L^p$ norms in this situation. This is achieved by using $L$-functions and their reciprocity formulae: certain special identities between two different moments of central values of $L$-functions.

## Moments of large families of Dirichlet $L$-functions

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.

## Discrete Moments

This talk aims to provide an overview of discrete moment computations, specifically, moments of objects related to the Riemann zeta-function when they are sampled at the nontrivial zeros of the zeta-function. We will discuss methods that have been used to do such calculations and will mention their applications.

- Read more about Discrete Moments
- 1052 reads

## Limitations to equidistribution in arithmetic progressions

It is well known that the prime numbers are equidistributed in arithmetic progressions. Such a phenomenon is also observed more generally for a class of arithmetic functions. A key result in this context is the Bombieri--Vinogradov theorem which establishes that the primes are equidistributed in arithmetic progressions ``on average" for moduli $q$ in the range $q\leq x^{1/2-\epsilon}$ for any $\epsilon > 0 $. Building on an idea of Maier, Friedlander--Granville showed that such equidistribution results fail if the range of the moduli $q$ is extended to $q\leq x/(\log x)^B$ for any $B>1$. We discuss variants of this result and give some applications. This is joint work with my supervisor Akshaa Vatwani

## Quantum variance for automorphic forms

In this talk, I will discuss the quantum variances for families of automorphic forms on modular surfaces. The resulting quadratic forms are compared with the classical variance. The proofs depend on moments of central $L$-values and estimates of the shifted convolution sums/non-split sums. (Based on joint work with Stephen Lester.)