Coefficients of cyclotomic polynomials

TL;DR

This paper extends previous results on the coefficients of cyclotomic polynomials, showing that for certain linear forms, all integers appear as coefficients across infinitely many polynomials.

Contribution

It generalizes earlier work by proving that for any integers s > t ≥ 0, the set of coefficients a(ns+t, k) covers all integers, broadening the understanding of cyclotomic polynomial coefficients.

Findings

The set of coefficients a(ns+t, k) equals all integers for s > t ≥ 0.

The result generalizes previous findings for multiples of n.

It deepens the understanding of the distribution of cyclotomic polynomial coefficients.

Abstract

Let $a(n, k)$ be the $k$-th coefficient of the $n$-th cyclotomic polynomial. Recently, Ji, Li and Moree \cite{JLM09} proved that for any integer $m\ge1$, $\{a(mn, k)| n, k\in\mathbb{N}\}=\mathbb{Z}$. In this paper, we improve this result and prove that for any integers $s>t\ge0$, $$\{a(ns+t, k)| n, k\in\mathbb{N}\}=\mathbb{Z}.$$