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1. Introduction. The aim of this paper is to supply a still better result for the problem considered in [2]. Let A(x) denote the number of distinct abelian groups (up to isomorphism) of orders not exceeding x. We shall proveTheorem 1. For any ε > 0,$A(x) = C₁x + C₂x^{1/2} + C₃x^{1/3} + O(x^{50/199+ε})$,where C₁, C₂ and C₃ are constants given on page 261 of [2].Note that 50/199=0.25125..., thus...
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