On a certain divisor function in Number fields

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...

Full description

Saved in:
Bibliographic Details
Published in:arXiv.org 2021-06
Main Authors: Gupta, Rajat, Pandit, Sudip
Format: Article
Language:English
Subjects:
Dirichlet problem
Mathematical functions
Number theory
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary: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\"ı summation formula associated to the divisor counting function \(d(n)\). We also obtain a big \(O\)-estimate for the Riesz sum associated to \(\sigma_{\mathbb{K},\alpha}(n)\).
ISSN:2331-8422