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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 UrbanaChampaign 
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 $n > 1$ $$\Phi\sb{n}(z)={\prod\limits\sb{d\vert n}}(1z\sp{d})\sp{\mu(n/d)},$$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 onesided 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:  20110507 
Identifier in Online Catalog:  AAI9210733 
OCLC Identifier:  (UMI)AAI9210733 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois