Hooley's delta function

In mathematics, Hooley's delta function, also called Erdős--Hooley delta-function, is 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 are

$1,2,1,2,1,2,1,2,1,2,1,3,1,2,2,2,1,2,1,3,2,2,1,4$ (sequence A226898 in the OEIS).

History

The sequence was first introduced by Paul Erdős in 1974, then studied by Christopher Hooley in 1979.

In 1985, Helmut Maier and Gérald Tenenbaum proved that $\textstyle \sum _{k=1}^{n}\Delta (k)<ne^{c{\sqrt {\log \log n\log \log \log n}}}$ for some constant $c>0$ and all $n>16$. In particular, the average order of $\Delta (n)$ is $O((\log n)^{k})$ for any $k>0$.

They also shows that ${\log \log n}^{g+O(1)}<\Delta (n)<{\log \log n}^{\log 2+O(1)}$ for almost all $n$, with $g={\log 2}/\log((1-1/\log 27)/(1-1/\log 3))\approx 0.338...$.

Usage

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

$\Delta (mn)\leq \Delta (n)d(m)$ where $d(n)$ is the number of divisors of $n$.

See also

• Divisor function
• Euler's number