Hooley's delta function

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

${\displaystyle 1,2,1,2,1,2,1,2,1,2,1,3,1,2,2,2,1,2,1,3,2,2,1,4}$ (sequence A226898 in the OEIS).

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

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

Later in 2023 Kevin Ford, Koukoulopoulos, and Tao proved the lower bound ${\displaystyle \textstyle \sum _{k=1}^{n}\Delta (k)\gg n(\log \log n)^{1+\eta -\epsilon }}$, where ${\displaystyle \eta =0.3533227\ldots }$, fixed ${\displaystyle \epsilon }$, and ${\displaystyle n\geq 100}$.^[5]

Usage

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

The growth of this sequence is limited by ${\displaystyle \Delta (mn)\leq \Delta (n)d(m)}$ where ${\displaystyle d(n)}$ is the number of divisors of ${\displaystyle n}$.^[6]

See also

• Divisor function
• Euler's number

References

1. Erdös, Paul (1974). "On Abundant-Like Numbers". Canadian Mathematical Bulletin. 17 (4): 599–602. doi:10.4153/CMB-1974-108-5. S2CID 124183643.
2. Hooley, Christopher. "On a new technique and its applications to the theory of numbers" (PDF). American Mathematical Society. Archived (PDF) from the original on 17 December 2022. Retrieved 17 December 2022.
3. Koukoulopoulos, D.; Tao, T. (2023). "An upper bound on the mean value of the Erdős–Hooley Delta function". Proceedings of the London Mathematical Society. 127 (6): 1865–1885. arXiv:2306.08615. doi: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". arXiv:2308.11987 [math.NT].
6. 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.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-18.