Number Theory

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.

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

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.