In this thesis we study the remainder term e(s) in the hyperbolic lattice point
counting problem. Our main approach to this problem is that of the spectral
theory of automorphic forms. We show that the function e(s) exhibits properties
similar to those of almost periodic functions, and we study dierent aspects of
the theory of almost periodic functions, namely criteria for the existence of
asymptotic moments and limiting distribution for such type of functions. This
gives us the possibility to infer nontrivial bounds on higher moments of e(s), and
existence of asymptotic moments and limiting distribution for certain integral
versions of it. Finally we describe what results can be obtained by application
of fractional calculus, especially fractional integration to small order, to the
problem.