ON THE ASYMPTOTIC-BEHAVIOR OF THE NUMBER OF SUBGROUPS OF FINITE ABELIAN-GROUPS

Let G congruent to Z/n(1)Zx...x Z/n(r)Z be a finite abelian group of r ank r, where n(j)\n(j+1) for j = 1,..., tau -1. Let tau(G) be the numb er of subgroups of G,\G\ the order of G and r(G) the rank of G. In thi s paper we investigate carefully the asymptotic behaviour of the level function l(tau)((r)) (n) := Sigma(\G\=n,r(G less than or equal to r)) for r = 2. In particular we prove that Sigma(n less than or equal to x) l(tau)((2)) (n)=A(1)x(logx)(2) + A(2)xlogx + A(3)x + Delta(x), wher e A(i)-s are constants, Delta(x) much less than X-5/8(logX)(4) and Del ta(x) = Omega-(X-1/2(logX)(2)).