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

Classification: 11N45, 11L07, 20K01, 20K27
@article{JTNB_1997__9_2_371_0,
author = {Ivi\'c, Aleksandar},
title = {On the number of subgroups of finite abelian groups},
journal = {Journal de th\'eorie des nombres de Bordeaux},
pages = {371--381},
publisher = {Universit\'e Bordeaux I},
volume = {9},
number = {2},
year = {1997},
mrnumber = {1617404},
zbl = {0905.11040},
language = {en},
url = {http://archive.numdam.org/item/JTNB_1997__9_2_371_0/}
}
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
SP  - 371
EP  - 381
VL  - 9
IS  - 2
PB  - Université Bordeaux I
UR  - http://archive.numdam.org/item/JTNB_1997__9_2_371_0/
LA  - en
ID  - JTNB_1997__9_2_371_0
ER  - 
%0 Journal Article
%A Ivić, Aleksandar
%T On the number of subgroups of finite abelian groups
%J Journal de théorie des nombres de Bordeaux
%D 1997
%P 371-381
%V 9
%N 2
%I Université Bordeaux I
%U http://archive.numdam.org/item/JTNB_1997__9_2_371_0/
%G en
%F JTNB_1997__9_2_371_0
Ivić, Aleksandar. On the number of subgroups of finite abelian groups. Journal de théorie des nombres de Bordeaux, Volume 9 (1997) no. 2, pp. 371-381. http://archive.numdam.org/item/JTNB_1997__9_2_371_0/

[1] G. Bhowmik, Average order of certain functions connected with arithmetic of matrices, J. Indian Math. Soc. 59 (1993), 97-106. | MR | Zbl

[2] G. Bhowmik and H. Menzer, On the number of subgroups of finite Abelian groups, Abh. Math. Sem. Univ. Hamburg, in press. | MR | Zbl

[3] G. Bhowmik and J. Wu, On the asymptotic behaviour of the number of subgroups of finite abelian groups, Archiv der Mathematik 69 (1997), 95-104. | MR | Zbl

[4] A. Ivi, The Riemann zeta-function, John Wiley & Sons, New York (1985). | MR | Zbl

[5] A. Ivić, The general divisor problem, J. Number Theory 27 (1987), 73-91. | MR | Zbl

[6] H.-Q. Liu, Divisor problems of 4 and 3 dimensions, Acta Arith. 73 (1995), 249-269. | MR | Zbl

[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. | Zbl

[8] K. Ramachandra, Progress towards a conjecture on the mean value of Titchmarsh series, Recent Progress in Analytic Number Theory, Academic Press, London 1 (1981), 303-318. | MR | Zbl

[9] K. Ramachandra, On the Mean- Value and Omega-Theorems for the Riemann zeta-function, Tata Institute of Fund. Research, Bombay, 1995. | MR | Zbl