Hooley's delta function

The sequence was first introduced by Paul Erdős in 1974,[1] then studied by Christopher Hooley in 1979.[2]

In 1985, Helmut Maier and Gérald Tenenbaum proved that ${\displaystyle \textstyle \sum _{k=1}^{n}\Delta (k)<ne^{c{\sqrt {\log \log n\log \log \log n}}}}$ for some constant ${\displaystyle c>0}$ and all ${\displaystyle n>16}$. In particular, the average order of ${\displaystyle \Delta (n)}$ is ${\displaystyle O((\log n)^{k})}$ for any ${\displaystyle k>0}$.[3][4]

They also shows that ${\displaystyle {\log \log n}^{g+O(1)}<\Delta (n)<{\log \log n}^{\log 2+O(1)}}$ for almost all ${\displaystyle n}$,[4] with ${\displaystyle g={\log 2}/\log((1-1/\log 27)/(1-1/\log 3))\approx 0.338...}$.[5]

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

${\displaystyle \Delta (mn)\leq \Delta (n)d(m)}$ where ${\displaystyle d(n)}$ is the number of divisors of ${\displaystyle n}$.[6]

• Divisor function
• Euler's number