# On the number of subgroups of finite abelian groups

• Volume: 9, Issue: 2, page 371-381
• ISSN: 1246-7405

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## Abstract

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

## How to cite

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Ivić, Aleksandar. "On the number of subgroups of finite abelian groups." Journal de théorie des nombres de Bordeaux 9.2 (1997): 371-381. <http://eudml.org/doc/248010>.

@article{Ivić1997,
abstract = {Let\begin\{equation*\} T(x) = K\_1x \log ^2 x + K\_2x \log x + K\_3x + \Delta (x), \end\{equation*\}where $T(x)$ denotes the number of subgroups of all abelian groups whose order does not exceed $x$ and whose rank does not exceed $2$, and $\Delta (x)$ is the error term. It is proved that\begin\{equation*\} \int \_1^X \Delta ^2(x) dx \ll X^2 \log ^\{31/3\} X, \int \_1^X \Delta ^2 ( x) dx = \Omega (X^2 \log ^4 X).\end\{equation*\}},
author = {Ivić, Aleksandar},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {number of subgroups of finite abelian groups; asymptotic results on counting functions for algebraic structures},
language = {eng},
number = {2},
pages = {371-381},
publisher = {Université Bordeaux I},
title = {On the number of subgroups of finite abelian groups},
url = {http://eudml.org/doc/248010},
volume = {9},
year = {1997},
}

TY - JOUR
AU - Ivić, Aleksandar
TI - On the number of subgroups of finite abelian groups
JO - Journal de théorie des nombres de Bordeaux
PY - 1997
PB - Université Bordeaux I
VL - 9
IS - 2
SP - 371
EP - 381
AB - Let\begin{equation*} T(x) = K_1x \log ^2 x + K_2x \log x + K_3x + \Delta (x), \end{equation*}where $T(x)$ denotes the number of subgroups of all abelian groups whose order does not exceed $x$ and whose rank does not exceed $2$, and $\Delta (x)$ is the error term. It is proved that\begin{equation*} \int _1^X \Delta ^2(x) dx \ll X^2 \log ^{31/3} X, \int _1^X \Delta ^2 ( x) dx = \Omega (X^2 \log ^4 X).\end{equation*}
LA - eng
KW - number of subgroups of finite abelian groups; asymptotic results on counting functions for algebraic structures
UR - http://eudml.org/doc/248010
ER -

## References

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6. [6] H.-Q. Liu, Divisor problems of 4 and 3 dimensions, Acta Arith.73 (1995), 249-269. Zbl0846.11056MR1364462
7. [7] H. Menzer, On the number of subgroups of finite Abelian groups, Proc. Conf. Analytic and Elementary Number Theory (Vienna, July 18-20, 1996), Universität Wien & Universität für Bodenkultur, Eds. W.G. Nowak and J. Schoißengeier, Wien (1996), 181-188. Zbl0879.11049
8. [8] K. Ramachandra, Progress towards a conjecture on the mean value of Titchmarsh series, Recent Progress in Analytic Number Theory, Academic Press, London1 (1981), 303-318. Zbl0465.10033MR637354
9. [9] K. Ramachandra, On the Mean- Value and Omega-Theorems for the Riemann zeta-function, Tata Institute of Fund. Research, Bombay, 1995. Zbl0845.11003MR1332493

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