  We first extend the multiplicativity property of arithmetic functions to the
setting of operators on the Fock space. Secondly, we use phase operators to get
representation of some extended arithmetic functions by operators on the Hardy
space. Finally, we show that radial limits to the boundary of the unit disc in
the Hardy space is useful in order to go back to the classical arithmetic
functions. Our approach can be understudied as a transition from the classical
number theory to quantum setting.
