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

{e-s)

x <L(x)<(e + e)

x. COROLLARY 1.8. (see also in [6]) lim L(x)

x = e. X->cO LEMMA 1.9. lim = 1. xln(x) PROOF. We use the following inequality, a form of Stirling's Theorem: x In(x) - x + — In(x) + n

^

2 ^ < In (x!) < y• In(x) - y + — ln(x) +1, 2 2 2 rr n 1 ln(x!) 1 , ln(x!) 1 ( 1 Hence 1 < —-—— < 1, and so —-—— = 1 + o . ln(x) xln(x) xln(x) ^^(x)^ THEOREM L lim , ln

(*

!

) = 1. ln(Z,(x))ln(x) PROOF. From Corollary 1.6. Lemma 1.9. we have ln(x!) ln(x!) ln(x!

) xln(x) _ *->«>xln(x) ^ln(/.(x))ln(x) ~^Tn(L(x)) ~ ln

(Z(x)) Using Stirling's Formula again and Corollary 1.6 we may obtain n as a limit Some other similar result for n was obtained in [3] and [4]. 69