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On the Coefficients of Cyclotomic Polynomials
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Memoirs of the American Mathematical Society
1993; 80 pp; softcover
Volume: 106
ISBN-10: 0-8218-2572-0
ISBN-13: 978-0-8218-2572-3
List Price: US$34 Individual Members: US$20.40
Institutional Members: US\$27.20
Order Code: MEMO/106/510

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.

Research mathematicians.

• 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