A generalization of Arai-Carlitz's identity
type: Item
instance of: scholarly article
title: A generalization of Arai-Carlitz's identity
zbMATH Open document ID: 1468.11219
published in: The Ramanujan Journal
publication date: 2021-05-06T00:00:00Z
review text: The authors of the paper present a generalization of the Arai-Carlitz's identity to the ring of algebraic integers concerning Dirichlet characters. The formulation of the general assertion is quite complex, but the general assertion implies the following interesting equality \[ \sum\limits_{a,b,a+b\in(\mathbb{Z}/n\mathbb{Z})^\times}\mathrm{gcd}(a+b-1,n)\chi(a)=\mu(d)\varphi(n_0^2/d)X(n/n_0)\tau(n/d) \] provided if \(n\) is positive integer and \(\chi\) is a Dirichlet character modulo \(n\) with conductor \(d\). Here \(n_0|n\) is such that \(n_0\) has the same prime factors with \(d\) and \(\mathrm{gcd}(n_0,n/n_0)=1\), \(\varphi\) denotes the Euler's totient function, \(\mu\) is the Möbius function, \(\tau\) is the divisor function, and \[ X(n)=\sum\limits_{a,b,a+b\in(\mathbb{Z}/n\mathbb{Z})^\times}1. \]
reviewed by: Jonas Šiaulys
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 11A25
zbMATH DE Number: 7345288
zbMATH Keywords: ring of algebraic integers
zbMATH Keywords: Dirichlet character
zbMATH Keywords: arithmetical sum
author: Yujie Wang
author: Chungang Ji
MaRDI profile type: Publication
full work available at URL: s11139-019-00236-y
OpenAlex ID: W3035349258
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DOI: 10.1007/S11139-019-00236-Y
description: scientific article; zbMATH DE number 7345288