On the Maximum of a Twisted Divisor Function

Given n∈ℕ and ξ∈ℝ, let τ(n;ξ)=∑d|ndiξ. Hall and Tenenbaum asked in their book \textit{Divisors} what is the value of maxξ∈[1,2]|τ(n;ξ)| 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 maxh∈[0,1]|ζ(1/2+it+ih)| when t is chosen uniformly at random from [0,T].