  Loop counting functions $U(x)$ estimate the number of "weighted" loops in a
digital representation of $x\in[-1,1]$. Roughly speaking, each $x$ is
considered as an infinite walk, where the steps of the walk correspond to
digits of $x$. The graph of loop counting functions $U$ has a fractal structure
that resembles complex mountain landscapes. In some sense, $U$ allows us to
look at random walks globally. These functions may be helpful in the analysis
of some hard problems related to the distribution of self-avoiding random walks
(SAW) in a multi-dimensional case since SAW closely relate to zeros of $U(x)$.
We note here that $U(x)$ can be naturally extended to a multidimensional
argument $x$. In this article, the focus will be on some analytic aspects. It
will be shown that integrals $\int x^AU(x)^Bdx$ with non-negative integers $A$
and $B$ can be expressed in terms of integrals of rational functions with
integer coefficients. Moreover, it will be shown that $\int x^A U(x)dx$ admits
closed-form expressions. Fourier series for $U$ is also computed. Finally, we
discuss some connections with special functions and generalized continued
fractions, and other perspectives.
