On the number of subgroups of finite abelian groups
Journal de Théorie des Nombres de Bordeaux, Volume 9 (1997) no. 2, pp. 371-381.

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 ${\int }_{1}^{X}{\Delta }^{2}\left(x\right)dx\ll {X}^{2}{log}^{31/3}X,{\int }_{1}^{X}{\Delta }^{2}\left(x\right)dx=\Omega \left({X}^{2}{log}^{4}X\right).$

Soit $T\left(x\right)={K}_{1}x{log}^{2}x+{K}_{2}xlogx+{K}_{3}x+\Delta \left(x\right),$$T\left(x\right)$ désigne le nombre de sous groupes des groupes abéliens dont l’ordre n’excède pas $x$ et dont le rang n’excède pas $2$, et $\Delta \left(x\right)$ est le terme d’erreur. On démontre que ${\int }_{1}^{X}{\Delta }^{2}\left(x\right)dx\ll {X}^{2}{log}^{31/3}X,{\int }_{1}^{X}{\Delta }^{2}\left(x\right)dx=\Omega \left({X}^{2}{log}^{4}X\right).$

DOI: 10.5802/jtnb.208
Classification: 11N45,  11L07,  20K01,  20K27
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title = {On the number of subgroups of finite abelian groups},
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Aleksandar Ivić. On the number of subgroups of finite abelian groups. Journal de Théorie des Nombres de Bordeaux, Volume 9 (1997) no. 2, pp. 371-381. doi : 10.5802/jtnb.208. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.208/

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