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

Let

T(x)=K 1 xlog 2 x+K 2 xlogx+K 3 x+Δ(x),
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 Δ(x) is the error term. It is proved that
∫ 1 X Δ 2 (x)dx≪X 2 log 31/3 X,∫ 1 X Δ 2 (x)dx=Ω(X 2 log 4 X).

Soit

T(x)=K 1 xlog 2 x+K 2 xlogx+K 3 x+Δ(x),
où T(x) 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 Δ(x) est le terme d’erreur. On démontre que
∫ 1 X Δ 2 (x)dx≪X 2 log 31/3 X,∫ 1 X Δ 2 (x)dx=Ω(X 2 log 4 X).

DOI : 10.5802/jtnb.208
Classification : 11N45, 11L07, 20K01, 20K27 
@article{JTNB_1997__9_2_371_0,
     author = {Aleksandar Ivi\'c},
     title = {On the number of subgroups of finite abelian groups},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {371--381},
     year = {1997},
     publisher = {Universit\'e Bordeaux I},
     volume = {9},
     number = {2},
     doi = {10.5802/jtnb.208},
     zbl = {0905.11040},
     mrnumber = {1617404},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.208/}
}
TY  - JOUR
AU  - Aleksandar Ivić
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  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.208/
DO  - 10.5802/jtnb.208
LA  - en
ID  - JTNB_1997__9_2_371_0
ER  - 
%0 Journal Article
%A Aleksandar Ivić
%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 https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.208/
%R 10.5802/jtnb.208
%G en
%F JTNB_1997__9_2_371_0
Aleksandar Ivić. On the number of subgroups of finite abelian groups. Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 371-381. doi: 10.5802/jtnb.208

[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

Cité par Sources :