#### by A. J. Hildebrand

Let
__\[
\Phi_n(z)=\prod_{\substack{r=1\\ (r,n)=1}}^{n}\!\!
\left(1-e^{2\pi i
r/n}\right)=\sum_{m=0}^{\phi(n)}a(m,n)z^m
\]__
be the __\( n \)__-th cyclotomic polynomial. In 1949, in his first published
paper, Paul Bateman
[1]
proved the inequality
__\begin{equation}
\max_{m}|a(m,n)|\le \exp\Big(\tfrac{1}{2}d(n)\log n\Big),
\label{one}
\end{equation}__
where __\( d(n) \)__ is the divisor function, and deduced from this
the bound
__\begin{equation}
\log \max_{m}|a(m,n)|\le
\exp\Bigl(\bigl(1+o(1)\bigr)\frac{(\log 2)(\log n)}{\log\log n}\Bigr).
\label{two}
\end{equation}__
Bateman’s proof of __\eqref{one}__ and __\eqref{two}__ takes just over one page, and is a model
of simplicity and elegance. His work was motivated by a 1946 paper of
Paul Erdős
[e1],
in which Erdős gave a lower bound for
__\( \max_m|a(m,n)| \)__ among other results and conjectures.

The papers by Erdős and Bateman gave impetus to significant further
work on the coefficients of cyclotomic polynomials. We mention here some
of the key results that have been obtained during the past few
decades. In 1974,
Robert Vaughan
[e2]
showed that the upper
bound __\eqref{two}__ is best possible, in the sense that an inequality in the
other direction holds for infinitely many __\( n \)__. In 1984, Bateman,
Carl Pomerance,
and Vaughan
[2]
gave upper and lower bounds for
__\( \max_m|a(m,n)| \)__ in terms of the number of prime factors of __\( n \)__. In 1990,
Helmut Maier
[e3]
proved a long-standing conjecture on
the “typical” size of the maximal coefficient __\( \max_m|a(m,n)| \)__ by
showing that, for “almost all” __\( n \)__, one has __\( \max_{m}|a(m,n)|\ge
n^{\epsilon(n)} \)__ for any given function __\( \epsilon(n)\to0 \)__. In 1993,
Gennady Bachman
[e4]
considered the size of the coefficient
__\( a(m,n) \)__ when __\( m \)__ is fixed and __\( n \)__ varies, and he obtained an
asymptotic formula for __\( \log \max_n|a(m,n)| \)__ as __\( m\to\infty \)__.

The above papers settle some of the key problems in this area, but many other open questions remain, and the subject continues to attract the attention of researchers. Of the fourteen papers that reference the 1984 paper [2] of Bateman, Pomerance, and Vaughan, according to the MathSciNet database, five have been published since 2010.