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

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).

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).

Classification : 11N45, 11L07, 20K01, 20K27 
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     title = {On the number of subgroups of finite abelian groups},
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     publisher = {Universit\'e Bordeaux I},
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     number = {2},
     year = {1997},
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     url = {http://www.numdam.org/item/JTNB_1997__9_2_371_0/}
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Ivić, Aleksandar. On the number of subgroups of finite abelian groups. Journal de théorie des nombres de Bordeaux, Tome 9 (1997) no. 2, pp. 371-381. http://www.numdam.org/item/JTNB_1997__9_2_371_0/

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