Integro-local and integral limit theorems on the large deviations of sums of random vectors: Regular distributions.

Borovkov, A.A.

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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 prove Theorem 1. For any ε > 0, $A (x) = C ₁ x + C ₂ x^{1 / 2} + C ₃ x^{1 / 3} + O (x^{50 / 199 + ε})$ , where C₁, C₂ and C₃ are constants given on page 261 of [2]. Note that 50/199=0.25125..., thus improving our previous exponent 40/159=0.25157... obtained in [2]. To prove Theorem 1, we shall proceed along the line of approach presented...