Nathan Creighton | Searcharxiv

arXiv2026

Lower bounds for the large deviations and moments of the Riemann zeta function on the critical line

Building on work in \cite{AB24} on the Riemann zeta function at height $T$ off the critical line, we prove an unconditional lower bound on the critical line for real large deviations of the order $V\sim\alpha\log\log T$ for any $\alpha>0.$ This gives another proof of the sharpest known unconditional lower bounds on the fractional moments of the Riemann zeta function, due to \cite{HSlower}. The lower bound on large deviations is of the same order of magnitude as the upper bound proved in \cite{AB23}, for the range $0<\alpha<2.$

math.NT↗

arXiv2024

Upper Bounds on Large Deviations of Dirichlet $L$-functions in the $q$-aspect

We prove a result on the large deviations of the central values of even primitive Dirichlet $L$-functions with a given modulus. For $V\sim \alpha\log\log q$ with $0<\alpha<1$, we show that \begin{equation}\nonumber\frac{1}{\varphi(q)} \# \left\{\chi \text{ even, primitive mod }q: \log \left|L\left(\chi,\frac{1}{2}\right)\right| >V\right\}\ll \frac{e^{-\frac{V^2}{\log\log q}}}{\sqrt{\log\log q}}.\end{equation} This yields the sharp upper bound for the fractional moments of central values of Dirichlet $L$-functions proved by Gao, upon noting that the number of even, primitive characters with modulus $q$ is $\frac{\varphi(q)}{2}+O(1).$ The proof is an adaptation to the $q$-aspect of the recursive scheme developed by Arguin, Bourgade and Radziwill for the local maxima of the Riemann zeta function, and applied by Arguin and Bailey to the large deviations in the $t$-aspect. We go further and get bounds on the case where $V=o(\log\log q)$. These bounds are not expected to be sharp, but the discrepancy from the Central Limit Theorem estimate grows very slowly with $q$. The method involves a formula for the twisted mollified second moment of central values of Dirichlet $L$-functions, building on the work of Iwaniec and Sarnak.

math.NT↗