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Let $\Delta(x)=\sum_{n\leq x}a(n)-\sum_{j=1}^6 c_jx^{1/j}$ denote the error term in the abelian group problem. Using zeta-function methods it is proved that $$\int_1^X\Delta^2(x)\,dx~&lt;\!\!&lt;~ X^{39/29} \log^2X$$ where the exponent $39/29=1.344827\ldots$ is close to the best possible exponent $4/3$ in this problem.

The number of finite non-isomorphic abelian groups in mean square.