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).

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

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 ${\displaystyle n\geq 100}$.

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 ${\displaystyle n}$.

• Divisor function
• Euler's number