**Memoirs of the American Mathematical Society**

1993;
80 pp;
Softcover

MSC: Primary 11;

Print ISBN: 978-0-8218-2572-3

Product Code: MEMO/106/510

List Price: $34.00

AMS Member Price: $20.40

MAA Member Price: $30.60

**Electronic ISBN: 978-1-4704-0087-3
Product Code: MEMO/106/510.E**

List Price: $34.00

AMS Member Price: $20.40

MAA Member Price: $30.60

# On the Coefficients of Cyclotomic Polynomials

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*Gennady Bachman*

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

# Table of Contents

## On the Coefficients of Cyclotomic Polynomials

- Contents v6 free
- 0. Introduction 18 free
- 1. Statement of results 411 free
- 2. Proof of Theorem 0; the upper bound 1118
- 3. Preliminaries 1320
- 4. Proof of Theorem 1; the minor arcs estimate 2835
- 5. Proof of Theorem 1; the major arcs estimate 3340
- 6. Proof of Theorem 2; preliminaries 5562
- 7. Proof of Theorem 2; completion 6471
- 8. Proof of Propositions 1 and 2 6875
- 9. Proof of Theorem 3 7077
- Appendix 7481
- References 7986