The Mean Value Of The Number Of Finite Abelian Groups In The Set Of Piatetski-Shapiro Prime

Let a(n) denote the number of non-isomorphic abelian groups with n elements. It is well known that a(n)is a positive,integer-valued and multiplicative function, with the property that a(pα)=P(α)for every prime p and every integerα≥1. P(α)is the number of unrestricted partition ofα,especially we have a(1)=1,a(p)=1,a(p2)=2,a(p3)=3, a(p4)=5,a(p5)=7,a(p6)=11,a(p7)=15. Many experts have studied the mean value of a(n) deeply: P.Erdos,G.Szekeres[1]first proved (?)=C1x+O(x1/2). Kendall,Rankin[2]proved the following result (?)=C1x+C2x1/2+O(x1/3logx). H.-E.Richert[3]proved the following result (?)=C1x+C2x1/2+C3x1/3+O(x3/10log9/10x). Letâ–³(x):=∑n≤x a(n)-C1x-C2x1/2-C3x1/3,the following are the latest results:â–³(x)《x20/69log21/23,W.Schwarz[4];â–³(x)《x7/27log2x,P.G.Schmidt[5];â–³(x)《x97/381log35x,G.Kolesnik[6]; â–³(x)《x40/159+ε,H.Q.Liu[7];â–³(x)《x50/199+ε,H.Q.Liu[7];â–³(x)《x55/219log7x,Sargos and Wu[8]；△(x)《x1/4+ε,Robert and Sargos[9].Suppose k≥2 is a fixed positive integer,standard decomposition of n＞1 is n=p1α1p2α2…psαs,nis called k-full number whenαj≥k(j=1,…,s).Letδk be characteristic function of k-full number set.Paper[18]prove the mean value of a(n)in 2-full and 3-full number set: where Pj(t)(j=1,2)is j power polynomial of t. where Qj(t)(j=2,4,6)is j power polynomial of t.1953,Piatetski-Shapiro[10] firstly considered the prime problem which is named by him.Let and he prove for 1＜c≤12/11≈1.090909….Since then,the range of c has been improved.The best result is Rivatå'ŒSargos[11],they prove the asymptotic formula(1)is right,for 1<c<2817/2426≈1.16117….The aim of the paper is to establish the following asymptotic formula for the mean value of a(n)in the sequence[nc]:Theoreml.1 Let A(x)=∑n≤xa([nc]),then for 1＜c＜2788/2396=1.16360606….