2026-09-23T10:50:34Z https://www.ideals.illinois.edu/oai-pmh

oai:www.ideals.illinois.edu:2142/23444 2023-07-10 col_2142_5131 col_2142_16340 com_2142_5130 com_2142_16339 com_2142_8903

Hildebrand, A.J. Bachman, Gennady 2011-05-07T14:14:24Z 2011-05-07T14:14:24Z 10000-01-01 1991 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. Made available in DSpace on 2011-05-07T14:14:24Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9210733.pdf: 2153806 bytes, checksum: 23e409ee3380d74f38ba8d419e1947ce (MD5) Previous issue date: 1991 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T15:04:31Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:30:50-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9210733 (UMI)AAI9210733 http://hdl.handle.net/2142/23444 eng Copyright 1991 Bachman, Gennady Mathematics On the coefficients of cyclotomic polynomials text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.