##
**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 c_{0} n(log n)^{-1} for an explicit constant c_{0} > 0. The authors show next that d (n!) /d ((n-1)!) = 1+P(n) n^{-1}+O(n^{- 1/2 }) where P(n) = **max**_{p | n}p. 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) < n^{4/9} 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

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag