Some Questions About Counting Functions Of Finite Abelian Groups

Let the arithmetical function a(n)denote the number on non-isomorphic abelian groups with n elements.It is well known that a(n)is a positive,integer-valued mul-tiplicative function,with the property that a(pα)=P(α)for every prime p and every integerα≥1.P(α)is the number of unrestricted partitions ofα,especially we have. Many experts have studied the mean value of a(n)deeply:P.Erdos,G.Szekeres first provedKendall,Rankin proved the following resultH.-E.Richert proved the following resultwe let△(x):=∑n≤sa(n)-c1x-c2x1/2-c3x1/3,The following are the latest results:△(x)《x20/69log21/23x,W.Schwarz；△(x)《x7/27log2x,P.G.Schmidt； △(x)《x97/381log35x,G.Kolesnik；△(x)《x40/159+ε,H.Q.Liu；△(x)《x50/199+ε,H.Q.Liu；△(x)《x55/219log7x,Sargos and Wu；△(x)《x1/4+ε,Robert and Sargos.In this paper,we want to discuss the mean value of a(n)in k-full number set.Suppose k≥2 is a fixed positive integer.Standard decomposition of n＞1 is n=p1α1…psαs.n is called k-full number whenαj≥k(j=1,…,s).We set 1 be k-full number.Letδk(n)express characteristic function of k-full number set.We have two Theorems in the following:Theorem 1:We have the asymptotic estimates .Where Pj(t)(j=1,2)is j power polynomial of t. Theorem 2 We have the asymptotic estimates Where Qj(t)(j=2,4,6)is j power polynomial of t.We will prove a hypothetical result:Theorem 3 Suppose Lindelof Hypothesis of Riemann-Zeta function establish, then Where HP(j)-1(t)is P(j)-1 power polynomial of t.