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

L The properties of the function L(x) For any positive integer x, L(x) can be written in the form a) £(*)=n p<

x Where k is defined for primes p by p

"<x<p

" and so for any prime p<x we have ln(x) (2) HP) where [ ] is the integer part function. From (1) and (2) (3) In (£(*)) = £ p<x ln(x) In (p) HP) follows, which was shown also in [6]. In the folio wings we prove some other properties of L(x). LEMMA 1.1. li

m In{L(x)

) _ i 6{x

) PROOF. By (3), using that k

p = 1 for p s with Vx < p < x, we get In(Z,(*))= £ k

p \n(p)+ X HP) = p<-Jx -Jx<p<x = 2 ln(/'-') + Ö(x). p<yfx But and X ln(/'-')<X ln(x) = ln(x)

• n( Vx) p<,Jx p< Jx 66