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

1 . I' '"(*) Vta(x) ( 3 ln(x) < 21n(x) x 1 + IK*

) ln(x)ri(x) < 1. 21n(x) The function n(x) can be approximated through the function L(x). We will show that H(x) is asymptotic to ln(Z(x))/ln(x) COROLLARY 1.5. \n(x)U(x) PROOF. From Lemmas 1.3 and 1.4, we have for x > 41, 11 < 0(x) ^ ln(Z,(x)

) ^ V ln(x) ln(x)ri(x) ln(x) TI(x) Now we can show that y/(x) - ln(L(x)) is asymptotic to x. COROLLARY 1.6. Ii

m In(L(x)

) = *->«> X PROOF. Using Lemma 1.3 and the inequalities (5) we get 1 fl(x) In(L(x)) n(x)ln(x) 3 I < s — s < i + •———. 21n(x) which x ln(x) x This actually shows that ln(/,(x)) = x + o implies the following two corollaries. ' x ^ ln(x)J COROLLARY 1.7. For all ^>0 and all sufficiently large x, 68