Mathematical function
In mathematics, Hooley's delta function (formula_1), also called Erdős--Hooley delta-function, defines the maximum number of divisors of formula_2 in formula_3 for all formula_4, where formula_5 is the Euler's number. The first few terms of this sequence are
formula_6 (sequence 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 the sum of the first formula_2 terms, formula_8, for formula_9. In particular, the average order of formula_1 to formula_11 is formula_12 for any formula_13.
Later in 2023 Kevin Ford, Koukoulopoulos, and Tao proved the lower bound formula_14, where formula_15, fixed formula_16, and formula_9.
Usage.
This function measures the tendency of divisors of a number to cluster.
The growth of this sequence is limited by formula_18 where formula_19 is the number of divisors of formula_20.
References.
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