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On the Coefficients of Cyclotomic Polynomials
 
Gennady Bachman University of Nevada, Las Vegas, NV
On the Coefficients of Cyclotomic Polynomials
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
On the Coefficients of Cyclotomic Polynomials
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On the Coefficients of Cyclotomic Polynomials
Gennady Bachman University of Nevada, Las Vegas, NV
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
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1061993; 80 pp
    MSC: 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.

    Readership

    Research 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1061993; 80 pp
MSC: 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.

Readership

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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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