## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  858.11051
Autor:  Erdös, Paul; Graham, S.W.; Ivic, Aleksandar; Pomerance, Carl
Title:  On the number of divisors of n! (In English)
Source:  Berndt, Bruce C. (ed.) et al., Analytic number theory. Vol. 1. Proceedings of a conference in honor of Heini Halberstam, May 16-20, 1995, Urbana, IL, USA. Boston, MA: Birkhäuser, Prog. Math. 138, 337-355 (1996).
Review:  In this interesting paper, various problems concerning the number of divisors of n! are investigated. The first theorem provides an asymptotic expansion for log d(n!) with first term c0 n(log n)-1 for an explicit constant c0 > 0. The authors show next that

d (n!) /d ((n-1)!) = 1+P(n) n-1+O(n- 1/2 )

where P(n) = maxp | np. This leads to an estimate for the least K = K(n) such that d((n+K)!) \geq 2d (n!). It follows that K(n)/ log n is unbounded but that K(n) < n4/9 for all sufficiently arge n. The final section concerns the difference D(n) = d(n!)-d((n-1)!). The authors call an integer n a champ of D(n) > D(m) whenever m < n. They show that p and 2p are champs for any prime p and conjecture that there are infinitely many champs not of this form.
Reviewer:  E.J.Scourfield (Egham)
Classif.:  * 11N37 Asymptotic results on arithmetic functions
05A10 Combinatorial functions
Keywords:  divisor functions; factorials; asymptotic results; champs

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