  The purpose of this paper is to give some explicit formulas involving
M\"obius functions, which may be known under the generalized Riemann
Hypothesis, but unconditional in this paper. Concretely, we prove explicit
formulas of partial sums of the M\"obius function in arithmetic progressions
and partial sums of the M\"obius functions on an Abelian number field $K$. In
addition, to obtain these explicit formulas, we study a certain finite Euler
product appearing from certain relation of primitive characters and imprimitive
characters in the present paper.
