Let $A(x)$ denote the number of isomorphism es of Abelian groups of order at most $x$. Then $A(x)=\sum _{j=1}^5c_jx^{1/j}+\Delta (x),$ with certain coefficients $c_j$. It is shown that $\int _0^X\Delta (x)^2dx\ll X^{4/3}(\log X)^{89},$ which is best possible, apart from the log power. The proof uses mean-value estimates for $\zeta (s)$. Using similar techniques it is shown that $\beta _5\leq \frac 9{20}$ in the usual notation of the generalized divisor problem. This result has been stated without proof by ZHANG [11].