Some explicit formulas for partial sums of Möbius functions

Abstract

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