On a certain divisor function in Number fields

TL;DR

This paper investigates an analogue of the divisor function in number fields, deriving formulas for Riesz sums, extending convergence results, and connecting to classical summation formulas like Voronoi's.

Contribution

It introduces a generalized divisor function in number fields, provides explicit Riesz sum formulas, and extends classical results to this broader setting.

Findings

Derived explicit Riesz sum formulas for the divisor function in number fields.

Extended convergence theorems for the Riesz sums.

Established a big O-estimate for the Riesz sums.

Abstract

The main aim of this paper is to study an analogue of the generalized divisor function in a number field $\mathbb{K}$, namely, $\sigma_{\mathbb{K},\alpha}(n)$. The Dirichlet series associated to this function is $\zeta_{\mathbb{K}}(s)\zeta_{\mathbb{K}}(s-\alpha)$. We give an expression for the Riesz sum associated to $\sigma_{\mathbb{K},\alpha}(n),$ and also extend the validity of this formula by using convergence theorems. As a special case, when $\mathbb{K}=\mathbb{Q}$, the Riesz sum formula for the generalized divisor function is obtained, which, in turn, for $\alpha=0$, gives the Vorono\"{\i} summation formula associated to the divisor counting function $d(n)$. We also obtain a big $O$-estimate for the Riesz sum associated to $\sigma_{\mathbb{K},\alpha}(n)$.