Values of coefficients of cyclotomic polynomials II

TL;DR

This paper extends previous results on the coefficients of cyclotomic polynomials, showing that for any integer m, the coefficients a(mn,k) can take on all integer values, using properties of reciprocal cyclotomic polynomials.

Contribution

It generalizes the earlier prime power case to arbitrary integers m, demonstrating the universality of coefficient values in cyclotomic polynomials.

Findings

a(mn,k) assumes every integer value for arbitrary m

Extension from prime power to general m

Uses properties of reciprocal cyclotomic polynomials

Abstract

Let a(n,k) be the kth coefficient of the nth cyclotomic polynomial. The first two authors showed in part I that if m is a prime power and n and k range over the non-negative integers, then a(mn,k) assumes every integer value. Here this result is extended to the case where m is arbitrary. The proof use some properties of reciprocal cyclotomic polynomials (see arXiv:0709.1570).