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(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)dxX 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),
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)dxX 2 log 31/3 X, 1 X Δ 2 (x)dx=Ω(X 2 log 4 X).

Classification: 11N45, 11L07, 20K01, 20K27
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     title = {On the number of subgroups of finite abelian groups},
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     pages = {371--381},
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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. https://jtnb.centre-mersenne.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