On next Thursday,
title: How concentrated can the divisors of a typical integer be?
talk by Dimitris Koukoulopoulos (Université de Montréal), October 15, Thursday, 2020, 14:00 Brazilian local time (GMT -3:00)
Abstract: The Delta function measures the concentration of the sequence of divisors of an integer. Specifically, given an integer $n$, we write $\Delta(n)$ for the maximum over $y$ of the number of divisors of $n$ lying in the dyadic interval $[y,2y]$. It was introduced by Hooley in 1979 because of its connections to various problems in Diophantine equations and approximation. In 1981, Maier and Tenenbaum proved that $\Delta(n)>1$ for almost all integers $n$, thus settling a 1948 conjecture due to Erd\H os. In subsequent work, they proved that $(\log\log n)^{c+o(1)}\le \Delta(n)\le (\log\log n)^{\log2+o(1)}$, where $c=(\log2)/\log(\frac{1-1/\log 27}{1-\log3})\approx 0.33827$ for almost all integers $n$. In addition, they conjectured that $\Delta(n)=(\log\log n)^{c+o(1)}$ for almost all $n$. In this talk, I will present joint work with Ben Green and Kevin Ford that disproves the Maier-Tenenbaum conjecture by replacing the constant $c$ in the lower bound by another constant $c'=0.35332277\dots$ that we believe is optimal. We also prove analogous results about permutations and polynomials over finite fields by reducing all three cases to an archetypal probabilistic model.
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