Some explicit formulas for partial sums of M\"obius functions

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.