Hooley's delta function

In mathematics, Hooley's delta function (

), also called Erdős--Hooley delta-function, defines the maximum number of divisors of

{\displaystyle [u,eu]}

is the Euler's number.

The first few terms of this sequence are The sequence was first introduced by Paul Erdős in 1974,[1] then studied by Christopher Hooley in 1979.

[2] In 2023, Dimitris Koukoulopoulos and Terence Tao proved that the sum of the first

terms,

( k ) ≪ n ( log ⁡ log ⁡ n

[3] In particular, the average order of

( ( log ⁡ n

[4] Later in 2023 Kevin Ford, Koukoulopoulos, and Tao proved the lower bound

( k ) ≫ n ( log ⁡ log ⁡ n

1 + η − ϵ

[5] This function measures the tendency of divisors of a number to cluster.

The growth of this sequence is limited by

is the number of divisors of