Number of Finite Non-isomorphic Abelian Groups in Short Intervals

Abstract

Let a(n) denote the number of non-isomorphic Abelian groups of order n. For afixed integer k≥1, letA_k(x, h):=sum from n=x<n≤x+h,a(n)=k to (1)If h≥x~(581/1744)logx=x~(0.33314…)logx as x→∞,it was proved by A,Ivic thatA_k(x, h)=(d_k+o(1))h, (1)whered_k=sum from n=1 to ∞ (1/2πn integral from n=-π to π(e~(ikt g_t(n)dt≥0))),g_t(n)=sum from n=d/n to (μ(n/d)e~(ita)(d)).In Ref. [2], A. Ivic and P. Shiu improved the result. They showed that if h≥x~(877/2653)(logx)~c=x~(0.3305…)(logx)~c,then Eq.(1)is true, where C is a computable constant. Based on the estimate for △(1, 2, 2;x) in Ref.[2] and elementary discussion, thisnote proves the following theorem, which gives an improvement to the problem.