On the Coefficients of Cyclotomic Polynomials

This book studies the coefficients of cyclotomic polynomials. Let $a(m,n)$ be the $m$ th coefficient of the $n$ th cyclotomic polynomial $Phi _n(z)$, and let $a(m)=mathrm{max}_n vert a(m,n)vert$. The principal result is an asymptotic formula for $mathrm{log}a(m)$ that improves a recent estimate of Montgomery and Vaughan. Bachman also gives similar formulae for the logarithms of the one-sided extrema $a^*(m)=mathrm{max}_na(m,n)$ and $a_*(m)=mathrm{ min}_na(m,n)$. In the course of the proof, estimates are obtained for certain exponential sums which are of independent interest.