  We deduce new properties of the orbicyclic function $E$ of several variables
investigated in a recent paper by V. A. Liskovets. We point out that the
function $E$ and its connection to the number of solutions of certain linear
congruences occur in the literature in a slightly different form. We
investigate another similar function considered by Deitmar, Koyama and Kurokawa
by studying analytic properties of some zeta functions of Igusa type. Simple
number theoretic proofs for some known properties are also given.
