ON THE WEIGHTED AVERAGE NUMBER OF CYCLIC SUBGROUPS OF Zl×Zm×Zn WITH lmn ⩽ x

Let Z n be the additive group of residue classes modulo n . For any positive integer l , m and n , let c ( l,m,n ) denote the total number of cyclic subgroups of the group Z l × Z m × Z n . Define D ~ 3 c : = ∑ lmn ⩽ x c ( l , m , n ) log x lmn . In this paper, we study the asymptotic behaviour of D ~ 3 c .