Hooley's delta function explained

Hooley's delta function
Named After:	Christopher Hooley
Publication Year:	1979
Author:	Paul Erdős
Oeis:	A226898
First Terms:	1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1

In mathematics, Hooley's delta function (

\Delta(n) ), also called Erdős--Hooley delta-function, defines the maximum number of divisors of n in [u, eu] for all u , where e is the Euler's number. The first few terms of this sequence are1 , 2 , 1 , 2 , 1 , 2 , 1 , 2 , 1 , 2 , 1 , 3 , 1 , 2 , 2 , 2 , 1 , 2 , 1 , 3 , 2 , 2 , 1 , 4 .

History

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

n terms, style \sum ^{n} _{k =1} \Delta(k) \ll n ( log log n) ^{11/4} , for n \ge 100 .^[3] In particular, the average order of \Delta(n) to k is O(( log n) ^{k)} for any k > 0 .^[4]

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

style \sum ^{n} _{k =1} \Delta(k) \gg n ( log log n) ^{1+η-\epsilon} , where η=0 . 3533227 \ldots , fixed \epsilon , and n \ge 100 .^[5]

Usage

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

The growth of this sequence is limited by

\Delta(mn) \leq \Delta(n) d(m) where d(n) is the number of divisors of n .^[6]

See also

• Divisor function
• Euler's number

Notes and References

1. 10.4153/CMB-1974-108-5. On Abundant-Like Numbers . 1974 . Erdös . Paul . Canadian Mathematical Bulletin . 17 . 4 . 599–602 . 124183643 . free .
2. Web site: Hooley . Christopher . On a new technique and its applications to the theory of numbers . live . https://web.archive.org/web/20221217084921/https://www.ams.org/journals/bull/1990-22-01/S0273-0979-1990-15871-9/S0273-0979-1990-15871-9.pdf . 17 December 2022 . 17 December 2022 . American Mathematical Society.
3. 2306.08615. An upper bound on the mean value of the Erdős–Hooley Delta function . 2023 . Koukoulopoulos . D. . Tao . T. . Proceedings of the London Mathematical Society . 127 . 6 . 1865–1885 . 10.1112/plms.12572 .
4. "O" stands for the Big O notation.
5. Ford . Kevin . Koukoulopoulos . Dimitris . Tao . Terence . 2023 . A lower bound on the mean value of the Erdős-Hooley Delta function . math.NT . 2308.11987.
6. Web site: Greathouse . Charles R. . Sequence A226898 (Hooley's Delta function: maximum number of divisors of n in [u, eu] for all u. (Here e is Euler's number 2.718... = A001113.)) ]. 2022-12-18 . . OEIS Foundation.