# Scientific

## 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.

## Lambert series of logarithm and a mean value theorem for $\zeta(\frac{1}{2}-it)\zeta'(\frac{1}{2}+it)$

## Negative moments of the Riemann zeta function

## The recipe for moments of $L$-functions

In 2005, Conrey, Farmer, Keating, Rubinstein, and Snaith formulated a `recipe' that leads to detailed conjectures for the asymptotic behavior of moments of various families of $L$-functions. In this talk, we will survey recent progress towards their conjectures and explore connections with different subjects.

## One-level density of zeros of Dirichlet L-functions over function fields

We compute the one-level density of zeros of order-$\ell$ Dirichlet $L$-functions over function fields $\mathbb{F}_q[t]$ for $\ell=3,4$ in the Kummer setting ($q\equiv1\pmod{\ell}$) and for $\ell=3,4,6$ in the non-Kummer setting ($q\not\equiv1\pmod{\ell}$). In each case, we obtain a main term predicted by Random Matrix Theory (RMT) and a lower order term not predicted by RMT. We also confirm the symmetry type of the family is unitary, supporting the Katz and Sarnak philosophy.