eBook ISBN:  9781470400873 
Product Code:  MEMO/106/510.E 
List Price:  $34.00 
MAA Member Price:  $30.60 
AMS Member Price:  $20.40 
eBook ISBN:  9781470400873 
Product Code:  MEMO/106/510.E 
List Price:  $34.00 
MAA Member Price:  $30.60 
AMS Member Price:  $20.40 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 106; 1993; 80 ppMSC: Primary 11
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 onesided 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.
ReadershipResearch mathematicians.

Table of Contents

Chapters

0. Introduction

1. Statement of results

2. Proof of Theorem 0; the upper bound

3. Preliminaries

4. Proof of Theorem 1; the minor arcs estimate

5. Proof of Theorem 1; the major arcs estimate

6. Proof of Theorem 2; preliminaries

7. Proof of Theorem 2; completion

8. Proof of Propositions 1 and 2

9. Proof of Theorem 3


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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 onesided 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.

Chapters

0. Introduction

1. Statement of results

2. Proof of Theorem 0; the upper bound

3. Preliminaries

4. Proof of Theorem 1; the minor arcs estimate

5. Proof of Theorem 1; the major arcs estimate

6. Proof of Theorem 2; preliminaries

7. Proof of Theorem 2; completion

8. Proof of Propositions 1 and 2

9. Proof of Theorem 3