The On-Line Encyclopedia of Integer Sequences (OEIS)

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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.)
(history; published version)

#101 by Charles R Greathouse IV at Tue Jun 27 08:09:40 EDT 2023

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#100 by Charles R Greathouse IV at Tue Jun 27 08:09:37 EDT 2023

FORMULA

The average order is between x log log x and x (log log x)^(11/4); the lower bound is due to Hall & Tenenbaum (1988) and the upper bound to Koukoulopoulos & Tao. - Charles R Greathouse IV, Jun 26 2023

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#99 by Michael De Vlieger at Mon Jun 26 18:07:48 EDT 2023

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proposed

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#98 by Charles R Greathouse IV at Mon Jun 26 16:56:45 EDT 2023

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proposed

#97 by Charles R Greathouse IV at Mon Jun 26 16:56:39 EDT 2023

REFERENCES

R. R. Hall and G. Tenenbaum, Divisors. Cambridge Tracts in Mathematics, 90. Cambridge University Press, Cambridge, 1988.

LINKS

Dimitris Koukoulopoulos and Terence Tao, <a href="https://arxiv.org/abs/2306.08615">A note on the mean value of the Erdős-Hooley Delta function</a>, arXiv preprint (2023). arXiv:2306.08615 [math.NT]

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#96 by Charles R Greathouse IV at Mon Dec 12 13:00:44 EST 2022

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#95 by Charles R Greathouse IV at Mon Dec 12 13:00:40 EST 2022

LINKS

Régis de la Bretèche and Gérald Tenenbaum, <a href="https://arxiv.org/abs/2210.13897">Two upper bounds for the Erdős--Hooley Delta-function</a>, arXiv preprint (2022). arXiv:2210.13897 [math.NT]

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#94 by Charles R Greathouse IV at Fri Jun 18 13:35:36 EDT 2021

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#93 by Charles R Greathouse IV at Fri Jun 18 13:35:30 EDT 2021

LINKS

Helmut Maier and Gérald Tenenbaum, <a href="httphttps://iecltenenb.perso.math.univ-lorrainecnrs.fr/~Gerald.Tenenbaum/PUBLIC/Prepublications_et_publicationsPPP/Delta(n)21.pdf">On the normal concentration of divisors</a>, J. London Math. Soc. 2 31:3 (1985), pp. 393-400.

Helmut Maier and Gérald Tenenbaum, <a href="httphttps://www.iecn.u-nancy.fr/~tenenb/PUBLIC.perso.math.cnrs.fr/Prepublications_et_publicationsPPP/Delta(n)2.pdf">On the normal concentration of divisors. II.</a>, Math. Proc. Cambridge Philos. Soc. 147:3 (2009), pp. 513-540.

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approved

editing

#92 by Charles R Greathouse IV at Fri Jun 18 13:28:49 EDT 2021

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editing

approved