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, $A\left(x\right)=C₁x+C₂x^{1/2}+C₃x^{1/3}+O\left(x^{50/199+\epsilon }\right)$, 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 in [2]....

Let$$T\left(x\right)=K_{1}x{log}^{2}x+K_{2}xlogx+K_{3}x+\Delta \left(x\right),$$where $T\left(x\right)$ denotes the number of subgroups of all abelian groups whose order does not exceed $x$ and whose rank does not exceed $2$, and $\Delta \left(x\right)$ is the error term. It is proved that$${\int }_1^X{\Delta }^2\left(x\right)dx\ll X^2{log}^{31/3}X,{\int }_1^X{\Delta }^2\left(x\right)dx=\Omega \left(X^2{log}^{4}X\right).$$

This article deals with the value distribution of multiplicative prime-independent arithmetic functions $\left(\alpha \right(n\left)\right)$ with $\alpha \left(n\right)=1$ if $n$ is $N$-free ($N\ge 2$ a fixed integer), $\alpha \left(n\right)>1$ else, and $\alpha \left(2^n\right)\to \infty$. An asymptotic result is established with an error term probably definitive on the basis of the present knowledge about the zeros of the zeta-function. Applications to the enumerative functions of Abelian groups and of semisimple rings of given finite order are discussed.