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

ln(x).n(Ví) -» 0 as x -> oo 0(x) from which the lemma follows. LEMMA 12. 0<n(*)ln(x)-ln U(x)) < 0(x) PROOF. From the inequality r - / < [r] ^ r

, using (3), we obtain n(x) ln(x) - 0(x) ^ ln(Z(x))g n(x)ln(x) and so the lemma is proved. LEMMA 1.3. 0(x)z ln(Z(x))^ n(*)ln(x). PROOF, p < x implies 1 < ln(x) /\n(p). Hence from (3) we obtain ln(x) 0(x) = ]>>(/>) g X p<x p<x IHP)} The other inequality is part of Lemma 1.2. We will need also the following result ln(p) = lnU(x)). LEMMA 1.4. m lim *->

c

o ln(*)n(*) 1. PROOF. We can deduce this from 6{x) « x plus the Prime Number Theorem (cf. e. g. in [2]). Or one can prove it from inequalities ' 1 A x / - x < 0(x\ ri(x) < (5) 1 ln(x) If ln(x)V 21n(x) for 41 < x, which can be found in [5], since by Lemma 1.3 we have 67