  This paper is associated with Nevanlinna class, Dirichlet series and
Szeg\"o's problem in infinitely many variables. As we will see, there is a
natural connection between these topics. The paper first introduces the
Nevanlinna class and the Smirnov class in this context, and generalizes the
classical theory in finitely many variables to the infinite-variable setting.
These results applied to Szeg\"o's problem on Hardy spaces in infinitely many
variables. Moreover, this paper is also devoted to the study of the
correspondence between the Nevanlinna functions and Dirichlet series.
