# Scientific

## $L^p$-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.

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## 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.)

## Logging of the zeta-function, but only for a few moments!

When we're between friends, we often throw in an $\epsilon$ here or there, and why not? Whether something grows like $(\log T)^{100}$ or just $T^{\epsilon}$ doesn?t often make much difference. I shall outline some current work, with Aleks Simoni\v{c}, on the error term in the fourth-moment of the Riemann zeta-function. We know that the $T^{\epsilon}$ in this problem can be replaced by a power of $\log T$ ? but which power? Tune in to find out.

## The third moment of quadratic $L$-Functions

I will present a smoothed asymptotic formula for the third moment of Dirichlet $L$-functions associated to real characters. Beyond the main term, which was known, the formula has an unexpected secondary term of size $x^{3/4}$ and an error of size $x^{2/3}$. I will give background on the multiple Dirichlet series techniques that motivated this result. And I will describe the new ideas about local and global multiple Dirichlet series that made the final, sieving step in the proof possible. This is joint work with Adrian Diaconu.

## The generalised Shanks's conjecture

## Asymptotic mean square of product of higher derivatives of the zeta-function and Dirichlet polynomials

We discuss the asymptotic behavior of the mean square of higher derivatives of the Riemann zeta function or Hardy's $Z$-function product with a Dirichlet polynomial in a short interval. As an application, we obtain a refinement of some results by Levinson--Montgomery as well as Ki--Lee on zero density estimates of higher derivatives of the Riemann zeta function near the critical line. Also, we obtain a zero distribution result for Matsumoto--Tanigawa's $\eta_k$-function. This is joint work with S. Pujahari.