In mathematics, Hooley's delta function, also called Erdős--Hooley delta-function, defines the maximum number of divisors of in for all , where is Euler's number. It is usually denoted . 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
for . In particular, the average order of to is for any .
In 2024, Kevin Ford along with Koukoulopoulos and Tao proved the lower bound
where is 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 .
See also Divisor function Euler's number
References