Special Year Research Seminar

Given $n\in\mathbb{N}$ and $\xi\in\mathbb{R}$, let $\tau(n;\xi)=\sum_{d|n}d^{i\xi}$. Hall and Tenenbaum asked in their book \textit{Divisors} what is the value of $\max_{\xi\in[1,2]} |\tau(n;\xi)|$ for a ``typical'' integer $n$. I will present work in progress, carried out in collaboration with Louis-Pierre Arguin and Paul Bourgarde, that answers this question. Our approach builds on the the recent work of Arguin, Belius, Bourgade, Radziwi\l\l~and Soundararajan about the distribution of $\max_{h\in[0,1]} |\zeta(1/2+it+ih)|$ when $t$ is chosen uniformly at random from $[0,T]$.