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

TL;DR

This paper derives explicit, unconditional formulas for partial sums of the M"obius function in arithmetic progressions and number fields, involving finite Euler products and character relations.

Contribution

It provides new explicit formulas for M"obius sums that do not rely on the generalized Riemann Hypothesis, expanding understanding of their behavior.

Findings

Explicit formulas for partial sums in arithmetic progressions

Explicit formulas for partial sums over Abelian number fields

Analysis of finite Euler products related to characters

Abstract

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.