Celebratio Mathematica

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.