摘要
Let a（n） denote the number of non-isomorphic Abelian groups of order n. For afixed integer k≥1, letA<sub>k</sub>（x, h）:=sum from n=x【n≤x+h,a（n）=k to （1）If h≥x<sup>581/1744</sup>logx=x<sup>0.33314…</sup>logx as x→∞,it was proved by A,Ivic thatA<sub>k</sub>（x, h）=（d<sub>k</sub>+o（1））h, （1）whered<sub>k</sub>=sum from n=1 to ∞ (1/2πn integral from n=-π to π(e<sup>ikt g<sub>t</sub>（n）dt≥0</sup>)),g<sub>t</sub>（n）=sum from n=d/n to (μ（n/d）e<sup>ita</sup>（d）).In Ref. [2], A. Ivic and P. Shiu improved the result. They showed that if h≥x<sup>877/2653</sup>（logx）<sup>c</sup>=x<sup>0.3305…</sup>（logx）<sup>c</sup>,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.
