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

hrj:117 - Hardy-Ramanujan Journal, January 1, 1986, Volume 9 - https://doi.org/10.46298/hrj.1986.117
The number of finite non-isomorphic abelian groups in mean square.

Authors: Aleksandar Ivić

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.


Volume: Volume 9
Published on: January 1, 1986
Submitted on: March 3, 2015
Keywords: number of non-isomorphic abelian groups,mean square estimates,power moments of the zeta-function.,[MATH] Mathematics [math]


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