eBook ISBN: | 978-1-4704-0087-3 |
Product Code: | MEMO/106/510.E |
List Price: | $34.00 |
MAA Member Price: | $30.60 |
AMS Member Price: | $20.40 |
eBook ISBN: | 978-1-4704-0087-3 |
Product Code: | MEMO/106/510.E |
List Price: | $34.00 |
MAA Member Price: | $30.60 |
AMS Member Price: | $20.40 |
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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 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.
ReadershipResearch mathematicians.
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Table of Contents
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Chapters
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0. Introduction
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1. Statement of results
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2. Proof of Theorem 0; the upper bound
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3. Preliminaries
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4. Proof of Theorem 1; the minor arcs estimate
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5. Proof of Theorem 1; the major arcs estimate
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6. Proof of Theorem 2; preliminaries
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7. Proof of Theorem 2; completion
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8. Proof of Propositions 1 and 2
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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 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.
-
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
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7. Proof of Theorem 2; completion
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8. Proof of Propositions 1 and 2
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9. Proof of Theorem 3