Hooley's delta function

In mathematics, Hooley's delta function (), also called Erdős--Hooley delta-function, defines the maximum number of divisors of in for all , where is the Euler's number. The first few terms of this sequence are

Hooley's delta function
Named afterChristopher Hooley
Publication year1979
Author of publicationPaul Erdős
First terms1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1
OEIS indexA226898
(sequence A226898 in the OEIS).

History

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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   terms,  , for  .[3] In particular, the average order of   to   is   for any  .[4]

Later in 2023 Kevin Ford, Koukoulopoulos, and Tao proved the lower bound  , where  , fixed  , and  .[5]

Usage

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

The growth of this sequence is limited by   where   is the number of divisors of  .[6]

See also

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References

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  1. ^ Erdös, Paul (1974). "On Abundant-Like Numbers". Canadian Mathematical Bulletin. 17 (4): 599–602. doi:10.4153/CMB-1974-108-5. S2CID 124183643.
  2. ^ Hooley, Christopher. "On a new technique and its applications to the theory of numbers" (PDF). American Mathematical Society. Archived (PDF) from the original on 17 December 2022. Retrieved 17 December 2022.
  3. ^ Koukoulopoulos, D.; Tao, T. (2023). "An upper bound on the mean value of the Erdős–Hooley Delta function". Proceedings of the London Mathematical Society. 127 (6): 1865–1885. arXiv:2306.08615. doi:10.1112/plms.12572.
  4. ^ "O" stands for the Big O notation.
  5. ^ Ford, Kevin; Koukoulopoulos, Dimitris; Tao, Terence (2023). "A lower bound on the mean value of the Erdős-Hooley Delta function". arXiv:2308.11987 [math.NT].
  6. ^ Greathouse, Charles R. "Sequence A226898 (Hooley's Delta function: maximum number of divisors of n in [u, eu] for all u. (Here e is Euler's number 2.718... = A001113.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-12-18.