On partial sums of the M\"obius and Liouville functions for number fields

TL;DR

This paper investigates the partial sums of the Möbius and Liouville functions in number fields, applying a new Perron's formula and exploring their connection to the grand Riemann hypothesis and prime ideal theorem.

Contribution

It introduces a novel application of Liu and Ye's Perron's formula to study these sums and examines their relation to key conjectures in algebraic number theory.

Findings

New bounds for partial sums of Möbius and Liouville functions

Connections established between these sums and the grand Riemann hypothesis

Insights into the distribution of prime ideals in number fields

Abstract

Landau examined the partial sums of the M\"obius function and the Liouville function for a number field $K$. First we shall try again the same problem by using a new Perron's formula due to Liu and Ye. Next we consider the equivalent theorem of the grand Riemann hypothesis for the Dedekind zeta-function of $K$ and that of the prime ideal theorem.