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On the Coefficients of Cyclotomic Polynomials
Gennady Bachman

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
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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.

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
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