Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1993. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 21)

James P. Jones and Péter Kiss: Properties of The Least Common Multiple Function

THEOREM 2. xl

2e

2

x (6) lim 7 r

—— = 71 — 21n(Z(x))x

2

x PFOOF. After we multiply the left side of (6) by 2 and take the log we obtain 2 In x

!+ 2x - 2x In x - In In L(x) = = 2In x

!+ 2x - 2xin(x) - In(x) - (In In L(x) - In x). The term ln(ln(L(x)))-ln(x) -> 0, as oo, by Corollary 1.6. One of the formulations of Stirling's Formula (cf. e. g. Artin [1]) says that there exists a S, such that I I c ln(x!) + x - xln(x) - —-ln(x) = — ln(2^) + —- (0<^<l). Multiply this by 2 and take tlie limit as x -> oo

, the theorem follows. EL Quotient after x

! is divided by L(x). We derive some induction results about the quotient x!/ L(x

), which is here denoted by Q(x). LEMMA 2.1. L(x +

1) = (l(x),x +

1) PROOF. Since L(x +1) = [l(x), x +1]. LEMMA 22. L(x+

1) divides L(x) (x + 1). PROOF. By Lemma 2.1. 70