  In the previous papers, we tried to analyze the complete loop counting
functions that count all the loops in an infinite random walk represented by
digits of a real number. In this paper, the consideration will be restricted to
the partial loop counting functions $V$ that count the returns to the origin
only. This simplification allows us to find closed-form expressions for various
integrals related to $V$. Some applications to the complete loop counting
functions, in particular, their connections with Bernoulli polynomials, are
also provided.
