On the Coefficients of Cyclotomic Polynomials

Studies the coefficients of cyclotomic polynomials. This book presents the principal result is an asymptotic formula for $/textnormal{log}a(m)$ that improves an estimate of Montgomery and Vaughan.

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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 $a(m)=/textnormal{max}_n /vert a(m,n)/vert$. The principal result is an asymptotic formula for $/textnormal{log}a(m)$ that improves a recent estimate of Montgomery and Vaughan. Bachman also gives similar formulae for the logarithms of the one-sided extrema $a^*(m)=/textnormal{max}_na(m,n)$ and $a_*(m)=/textnormal{min}_na(m,n)$. In the course of the proof, estimates are obtained for certain exponential sums which are of independent interest.