On the number of cyclic subgroups of a finite abelian group
  We prove by using simple number-theoretic arguments formulae concerning the number of elements of a fixed order and the number of cyclic subgroups of a direct product of several finite cyclic groups. We point out that certain multiplicative properties of related counting functions for finite Abelian groups are immediate consequences of these formulae. 
L\'aszl\'o T\'oth
math.GR math.NT