Cyclotomic polynomial coefficients \(a(n,k)\) with \(n\) and \(k\) in prescribed residue classes
type: Item
instance of: scholarly article
title: Cyclotomic polynomial coefficients \(a(n,k)\) with \(n\) and \(k\) in prescribed residue classes
zbMATH Open document ID: 1253.11041
author: Jessica Fintzen
published in: Journal of Number Theory
publication date: 2011-07-22T00:00:00Z
review text: Let \(n\geq 1\) be an integer. Let \(\Phi_n(x)=\sum_{k=0}^{\varphi(n)} a(n,k) x^k\) denote the \(n\)th cyclotomic polynomial. Set \(a(n,k)=0\) for \(k>\varphi(n)\). The coefficients \(a(n,k)\) have a strong tendency to be small, e.g., for \(n<105\) one has \(a(n,k)\in \{-1,0,1\}\). Nevertheless it can be shown that \textit{every} integer can occur as a cyclotomic coefficient. In this paper the author determines the set \(\{a(n,k)\}\) as \(n\) runs through the positive integers congruent to \(a\pmod d\) and \(k\) runs through the non-negative integers congruent to \(b\pmod f\), with \(0\leq a<d\) and \(0\leq b<f\). Denote this set by \(S(a,d;b,f)\). Define the squarefree integer \(t(n)\) to be the product of the distinct primes dividing \(n\), and set \(s(n)=n/t(n)\). The author's main result is that if \(\gcd(s(\text{gcd}(a,d)),f)\) divides \(b\), then \(S(a,d;b,f)=\mathbb Z\) and \(S(a,d;b,f)=\{0\}\) otherwise. The proof uses the quantitative form of Dirichlet's prime number theorem for arithmetic progressions. \textit{C. G. Ji, W.-P. Li} and the reviewer [Discrete Math. 309, N. 6, 1720--1723 (2009; Zbl 1221.11067)] proved that \(S(0,d;b,1)=\mathbb Z\), a special case of the author's result. Consider the power series expansion of \(\Phi_n(x)^{-1}\) at \(x=0\): \(\Phi_n(x)^{-1}=\sum_{k=0}^{\infty}c(n,k)x^k\). It is easy to see that the \(c(n,k)\) are integers. The author also shows that her main result remains true if one replaces \(a(n,k)\) by \(c(n,k)\). For a detailed study of the coefficients \(c(n,k)\) see the reviewer [J. Number Theory 129, No. 3, 667--680 (2009; Zbl 1220.11037)].
reviewed by: Pieter Moree
Mathematics Subject Classification ID: 11C08
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 5931159
zbMATH Keywords: cyclotomic polynomial
zbMATH Keywords: Dirichlet's theorem
zbMATH Keywords: residue classes
MaRDI profile type: Publication
full work available at URL: j.jnt.2011.03.012
OpenAlex ID: W1976826246
cites work: Value distribution of cyclotomic polynomial coefficients
cites work: Values of coefficients of cyclotomic polynomials. II
cites work: On the magnitude of the coefficients of the cyclotomic polynomial
cites work: Inverse cyclotomic polynomials
cites work: On coefficients of cyclotomic polynomials
cites work: Q4454822
DOI: 10.1016/J.JNT.2011.03.012
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description: scientific article; zbMATH DE number 5931159