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Hooley's delta function

Inmathematics, 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).

History[edit]

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 ${\displaystyle n}$ terms, ${\displaystyle \textstyle \sum _{k=1}^{n}\Delta ($k)\ll n(\log \log n)^{11/4}} , for ${\displaystyle n\geq 100}$.^[3] In particular, the average orderof${\displaystyle \Delta ($n)} to${\displaystyle k}$is${\displaystyle O((\log n)^{k})}$ for any ${\displaystyle k>0}$.^[4]

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 ${\displaystyle n\geq 100}$.^[5]

Usage[edit]

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 divisorsof${\displaystyle n}$.^[6]

See also[edit]

• Divisor function
• Euler's number

References[edit]