Hooley's delta function

Hooley's delta function

Named after	Christopher Hooley
Publication year	1979
Author of publication	Paul Erdős
First terms	1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1
OEIS index	A226898

Short description: Mathematical function

In mathematics, Hooley's delta function ([math]\displaystyle{ \Delta(n) }[/math]), also called Erdős--Hooley delta-function, defines the maximum number of divisors of [math]\displaystyle{ n }[/math] in [math]\displaystyle{ [u, eu] }[/math] for all [math]\displaystyle{ u }[/math], where [math]\displaystyle{ e }[/math] is the Euler's number. The first few terms of this sequence are

[math]\displaystyle{ 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 3, 2, 2, 1, 4 }[/math] (sequence A226898 in the OEIS).

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 [math]\displaystyle{ n }[/math] terms, [math]\displaystyle{ \textstyle \sum_{k=1}^n \Delta(k) \ll n (\log \log n)^{11/4} }[/math], for [math]\displaystyle{ n \ge 100 }[/math]. In particular, the average order of [math]\displaystyle{ \Delta(n) }[/math] to [math]\displaystyle{ k }[/math] is [math]\displaystyle{ O((\log n)^k) }[/math] for any [math]\displaystyle{ k \gt 0 }[/math].

Later in 2023 Kevin Ford, Koukoulopoulos, and Tao proved the lower bound [math]\displaystyle{ \textstyle \sum_{k=1}^n \Delta(k) \gg n (\log \log n)^{1+\eta-\epsilon} }[/math], where [math]\displaystyle{ \eta=0.3533227\ldots }[/math], fixed [math]\displaystyle{ \epsilon }[/math], and [math]\displaystyle{ n \ge 100 }[/math].

Usage

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

The growth of this sequence is limited by [math]\displaystyle{ \Delta(mn) \leq \Delta(n) d(m) }[/math] where [math]\displaystyle{ d(n) }[/math] is the number of divisors of [math]\displaystyle{ n }[/math].

See also

• Divisor function
• Euler's number