This book studies the coefficients of cyclotomic polynomials. Let \(a(m,n)\) be the \(m\) th coefficient of the \(n\) th cyclotomic polynomial \(\Phi _n(z)\), and let \(a(m)=\mathrm{max}_n \vert a(m,n)\vert\). The principal result is an asymptotic formula for \(\mathrm{log}a(m)\) that improves a recent estimate of Montgomery and Vaughan. Bachman also gives similar formulae for the logarithms of the one-sided extrema \(a^*(m)=\mathrm{max}_na(m,n)\) and \(a_*(m)=\mathrm{ min}_na(m,n)\). In the course of the proof, estimates are obtained for certain exponential sums which are of independent interest.

Readership

Research mathematicians.

Table of Contents

• Introduction
• Statement of results
• Proof of Theorem 0; upper bound
• Preliminaries
• Proof of Theorem 1; the minor arcs estimate
• Proof of Theorem 1; the major arcs estimate
• Proof of Theorem 2; preliminaries
• Proof of Theorem 2; completion
• Proof of Propositions 1 and 2
• Proof of Theorem 3
• Appendix
• References