On a disctribution property of the residual order of \(a\pmod p\)
type: Item
instance of: scholarly article
title: On a disctribution property of the residual order of \(a\pmod p\)
zbMATH Open document ID: 1071.11054
author: Koji Chinen
author: Leo Murata
published in: Proceedings of the Japan Academy. Series A
publication date: 2003-01-01T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1116443604
review text: Let \(a\) be a rational number and let \(Q_a(x,k,l)\) be the set of primes \(p\leq x\) such that the (residual) order of \(a(\text{ mod~}p)\) is congruent to \(l(\text{ mod~}k)\). In case \(a\) is a positive integer which is not a perfect \(h\)th-power with \(h\geq 2\) and \(k=4\) the authors announce various results regarding \(Q_a(x,k,l)\) and sketch an outline of the proofs. Meanwhile these results and complete proofs have appeared in [J. Number Theory 105, 60--81 (2004; Zbl 1045.11066) and J. Number Theory 105, 82--100 (2004; Zbl 1045.11067)]. Of the two main results Theorem 1.1 is not new (see Zbl 1045.11066 for details). \textit{D. Zagier} (personal communication) was the first to find a much shorter reformulation of Theorem 1.2. In the reviewer's approach such a compact formulation results directly. Moreover, he extended Theorem 1.2 to arbitrary rational \(a\) and with smaller error term [J. Number Theory 114, 238--271 (2005; Zbl 1099.11052)].
reviewed by: Pieter Moree
Mathematics Subject Classification ID: 11N05
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11R18
zbMATH DE Number: 2024715
zbMATH Keywords: order
zbMATH Keywords: natural density
zbMATH Keywords: residue classes
MaRDI profile type: Publication
OpenAlex ID: W4251108728
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DOI: 10.3792/PJAA.79.28
description: scientific article; zbMATH DE number 2024715