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

Abstract : 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 $$\int_1^X\Delta^2(x)\,dx~<\!\!<~ X^{39/29} \log^2X$$ where the exponent $39/29=1.344827\ldots$ is close to the best possible exponent $4/3$ in this problem.
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Aleksandar Ivić. The number of finite non-isomorphic abelian groups in mean square.. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 1986, Volume 9 -1986, pp.17-23. ⟨10.46298/hrj.1986.117⟩. ⟨hal-01104704⟩

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