Hooley's delta function

In mathematics, Hooley's delta function, 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 Euler's number. It is usually denoted ${\displaystyle \Delta (n)}$. The first few terms of this sequence are

${\displaystyle 1,2,1,2,1,2,1,2,1,2,1,3,1,2,2,2,1,2,1,3,2,2,1,4,\dots }$

(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

${\displaystyle \sum _{k=1}^{n}\Delta (k)\ll n(\log \log n)^{11/4}}$

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 ${\displaystyle k>0}$.

In 2024, Kevin Ford along with Koukoulopoulos and Tao proved the lower bound

${\displaystyle \sum _{k=1}^{n}\Delta (k)\gg n(\log \log n)^{1+\eta -\epsilon },}$

where ${\displaystyle \epsilon }$ is fixed, ${\displaystyle \eta =0.3533227\ldots }$, and ${\displaystyle n\geq 100}$.

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 ${\displaystyle n}$.

See also

• Divisor function
• Euler's number

References