Editing Cyclotomic polynomial (section)

===Integers appearing as coefficients===

The problem of bounding the magnitude of the coefficients of the cyclotomic polynomials has been the object of a number of research papers.<ref name=arXivSanna>{{cite arXiv|eprint=2111.04034 |last1=Sanna |first1=Carlo |title=A Survey on Coefficients of Cyclotomic Polynomials |year=2021 |class=math.NT|mode=cs2 }}</ref>
 
If ''n'' has at most two distinct odd prime factors, then Migotti showed that the coefficients of <math>\Phi_n</math> are all in the set {1, −1, 0}.<ref>{{citation |title=Algebra: A Graduate Course |first=Martin |last=Isaacs |page=310 |isbn=978-0-8218-4799-2 |publisher=AMS Bookstore |year=2009}}</ref>

The first cyclotomic polynomial for a product of three different odd prime factors is <math>\Phi_{105}(x);</math> it has a coefficient −2 (see [[#Examples|above]]). The converse is not true: <math>\Phi_{231}(x)=\Phi_{3\times 7\times 11}(x)</math> only has coefficients in {1, −1, 0}.

If ''n'' is a product of more different odd prime factors, the coefficients may increase to very high values. E.g., <math>\Phi_{15015}(x) =\Phi_{3\times 5\times 7\times 11\times 13}(x)</math> has coefficients running from −22 to 23; also <math>\Phi_{255255}(x)=\Phi_{3\times 5\times 7\times 11\times 13\times 17}(x)</math>, the smallest ''n'' with 6 different odd primes, has coefficients of magnitude up to 532.

Let ''A''(''n'') denote the maximum absolute value of the coefficients of <math>\Phi_{n}(x)</math>.  It is known that for any positive ''k'', the number of ''n'' up to ''x'' with ''A''(''n'') > ''n''<sup>''k''</sup> is at least ''c''(''k'')⋅''x'' for a positive ''c''(''k'') depending on ''k'' and ''x'' sufficiently large.  In the opposite direction, for any function ψ(''n'') tending to [[infinity]] with ''n'' we have ''A''(''n'') bounded above by ''n''<sup>ψ(''n'')</sup> for almost all ''n''.<ref name=Mai2008>{{Citation
 | last = Maier
 | first = Helmut
 | chapter = Anatomy of integers and cyclotomic polynomials
 | editor1-last = De Koninck
 | editor1-first = Jean-Marie
 | editor2-last = Granville
 | editor2-first = Andrew
 | editor2-link = Andrew Granville
 | editor3-last = Luca
 | editor3-first = Florian
 | title = Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13-17, 2006
 | location = Providence, RI
 | publisher = [[American Mathematical Society]]
 | series = CRM Proceedings and Lecture Notes
 | volume = 46
 | pages = 89–95
 | year = 2008
 | isbn = 978-0-8218-4406-9
 | zbl = 1186.11010
}}</ref>

A combination of theorems of Bateman and Vaughan states that{{r|arXivSanna|p=10}} on the one hand, for every <math>\varepsilon>0</math>, we have
:<math>A(n) < e^{\left(n^{(\log 2+\varepsilon)/(\log\log n)}\right)}</math>
for all sufficiently large positive integers <math>n</math>, and on the other hand, we have
:<math>A(n) > e^{\left(n^{(\log 2)/(\log\log n)}\right)}</math>
for infinitely many positive integers <math>n</math>. This implies in particular that [[univariate polynomial|univariate polynomials]] (concretely <math>x^n-1</math> for infinitely many positive integers <math>n</math>) can have factors (like <math>\Phi_n</math>) whose coefficients are [[Superpolynomial|superpolynomially]] larger than the original coefficients. This is not too far from the general [[Landau-Mignotte bound]].