Given n∈ℕ and ξ∈ℝ, let τ(n;ξ)=∑d|ndiξ. Hall and Tenenbaum asked in their book 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, Radziwiłł, and Soundararajan about the distribution of maxh∈[0,1] |ζ(1/2+it+ih)| when t is chosen uniformly at random from [0,T].
This video is part of the Institute for Advanced Study‘s Special year research seminar.
