Hooley's delta function

In mathematics,[Average order of an arithmetic function|] Hooley's delta function , also called Erdős--Hooley delta-function, defines the maximum number of divisors of in for all, where is the Euler's number. The first few terms of this sequence are

History

The sequence was first introduced by Paul Erdős in 1974, then studied by Christopher Hooley in 1979.
In 2023, Dimitris Koukoulopoulos and Terence Tao proved that the sum of the first terms,, for. In particular, the
average order of to is for any.
Later in 2023 Kevin Ford, Koukoulopoulos, and Tao proved the lower bound, where, fixed, and.

Usage

This function measures the tendency of divisors of a number to cluster.
The growth of this sequence is limited by where is the number of divisors of.