Mean values and moments of arithmetic functions over number fields

TL;DR

This paper extends the understanding of mean values and moments of arithmetic functions, specifically the divisor function, over multiple number fields of odd degree with coprime discriminants, using adapted analytical methods.

Contribution

It generalizes previous results to finitely many number fields of odd degree with pairwise coprime discriminants and analyzes their combined divisor sum behavior.

Findings

Derived asymptotic formulas for mean values over multiple number fields.

Established moments of the error term in these asymptotics.

Analyzed the combined summatory function of divisor functions across two number fields.

Abstract

For an odd integer $d > 1$ and a finite Galois extension $K/\mathbb{Q}$ of degree $d$, G. L\"{u} and Z. Yang \cite{lu3} obtained an asymptotic formula for the mean values of the divisor function for $K$ over square integers. In this article, we obtain the same for finitely many number fields of odd degree and pairwise coprime discriminants, together with the moment of the error term arising here, following the method adapted by S. Shi in \cite{shi}. We also define the sum of divisor function over number fields and find the asymptotic behaviour of the summatory function of two number fields taken together.