Abstracts for Conference on "Progress and Emergent Theories in Zeta and L-Functions" (PRETZL)

This is joint with David Farmer, Chung Hang Kwan, Yongxiao Lin, and Caroline Turnage-Butterbaugh.
When studying the zeros of Riemann zeta function at a height $T$ up the critical strip one often multiplies
$\zeta$ times a Dirichlet polynomial, called a mollifier, of length $T^{\theta}$ before averaging in order to
pacify the irregularities of $\zeta$. The $\theta$ parameter here is critical.
Farmer conjectured that the mean square formulas one obtains for mollified zeta for small $\theta$ actually

Selbergs Central Limit Theorem asserts that the typical value of the logarithm of the zeta function at a height $T$
on the critical line is normally distributed with a standard deviation of order $(\log \log T)^{1/2}$. In this talk, we
will discuss recent works showing that these normal fluctuations persist, up to a constant, for values of the order of
the variance $\log \log T$. The results naturally relate to the works of Soundararajan and Harper on sharp upper
bounds of the $2k$-moments, as well as the work of Heap and Soundararajan on lower bounds. The connections

The construction and estimation of appropriate cubic moments of $L$-functions in small families
has been responsible for the strongest-known bounds for degree $2$ automorphic $L$-functions.
The Weyl bound for all Dirichlet $L$-functions is a consequence of this line of work. In previous
work with Petrow, we constructed families of twisted $L$-functions which may be interpreted as
capturing the $L$-functions whose underlying automorphic representation is everywhere principal