Английская Википедия:Hooley's delta function
Шаблон:Infobox integer sequenceШаблон:Short description In mathematics, Hooley's delta function (<math> \Delta(n) </math>), also called Erdős--Hooley delta-function, defines the maximum number of divisors of <math> n </math> in <math> [u, eu] </math> for all <math> u </math>, where <math> e </math> is the Euler's number. The first few terms of this sequence are
- <math>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> Шаблон:OEIS.
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 <math> n </math> terms, <math> \textstyle \sum_{k=1}^n \Delta(k) \ll n (\log \log n)^{11/4} </math>, for <math>n \ge 100</math>.[3] In particular, the average order of <math> \Delta(n) </math> to <math> k </math> is <math> O((\log n)^k) </math> for any <math> k > 0 </math>.[4]
Later in 2023 Kevin Ford, Koukoulopoulos, and Tao proved the lower bound <math> \textstyle \sum_{k=1}^n \Delta(k) \gg n (\log \log n)^{1+\eta-\epsilon} </math>, where <math>\eta=0.3533227\ldots</math>, fixed <math>\epsilon</math>, and <math>n \ge 100</math>.[5]
Usage
This function measures the tendency of divisors of a number to cluster.
The growth of this sequence is limited by <math>\Delta(mn) \leq \Delta(n) d(m)</math> where <math>d(n)</math> is the number of divisors of <math>n</math>.[6]
See also
References
Шаблон:Reflist Шаблон:Classes of natural numbers
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite web
- ↑ Шаблон:Cite journal
- ↑ "O" stands for the Big O notation.
- ↑ Шаблон:Cite arXiv
- ↑ Шаблон:Cite web
