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Title:On the coefficients of cyclotomic polynomials
Author(s):Bachman, Gennady
Doctoral Committee Chair(s):Hildebrand, A.J.
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Mathematics
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.
Issue Date:1991
Type:Text
Language:English
URI:http://hdl.handle.net/2142/23444
Rights Information:Copyright 1991 Bachman, Gennady
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9210733
OCLC Identifier:(UMI)AAI9210733


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