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https://hdl.handle.net/2142/23444
Description
Title
On the coefficients of cyclotomic polynomials
Author(s)
Bachman, Gennady
Issue Date
1991
Doctoral Committee Chair(s)
Hildebrand, A.J.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
Abstract
Let $\Phi\sb{n}(z)$ denote the $n$th cyclotomic polynomial, given by$$\Phi\sb{n}(z) = {\prod\limits\sbsp{a=1\atop(a,n)=1}{n}}\ (z - \exp(2\pi ia/n)) = {\sum\limits\sbsp{m=0}{\phi(n)}} a(m,n)z\sp{m}.$$It is easily verified that for $n > 1$ $$\Phi\sb{n}(z)={\prod\limits\sb{d\vert n}}(1-z\sp{d})\sp{\mu(n/d)},$$where $\mu$ is the Moebius function. Hence the coefficients $a(m,n)$ of $\Phi\sb{n}(z)$ are integers, and for every fixed $m,\ a(m,n)$ assumes only finitely many possible values.
We consider here the behavior of the function$$a(m) = {\max\limits\sb{n}}\ \vert a(m,n)\vert.$$Our principal result is an asymptotic formula for log $a(m)$ with logarithmic error term that improves over a recent estimate of Montgomery and Vaughan. We also give similar formulae for the logarithms of the one-sided extrema $a\sp* (m)$ = max$\sb{n}\ a(m,n)$ and $a\sb*(m)$ = min$\sb{n}\ a(m,n).$ In the course of the proof we obtain estimates for certain exponential sums which are of independent interest.
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