# Department of Pure Mathematics and Mathematical Statistics

Let n be a random integer (sampled from {1,..,X} for some large
X). It is a classical fact that, typically, n will have around (log n)^{log
2} divisors. Must some of these be close together? Hooley's Delta function
Delta(n) is the maximum, over all dyadic intervals I = [t,2t], of the
number of divisors of n in I. I will report on joint work with Kevin Ford
and Dimitris Koukoulopoulos where we conjecture that typically Delta(n) is
about (log log n)^c for some c = 0.353.... given by an equation involving
an exotic recurrence relation, and then prove (in some sense) half of this
conjecture, establishing that Delta(n) is at least this big almost surely.
For the most part I will discuss a model combinatorial problem about
representing integers in many ways as sums of elements from a random set.

19May
May 19th 2022
14:30 to 15:30

MR12

## Speaker:

Ben Green (Oxford)

## Series:

Combinatorics Seminar