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The coefficients of cyclotomic polynomials. (English) Zbl 0719.11012
Analytic number theory, Proc. Conf. in Honor of Paul T. Bateman, Urbana/IL (USA) 1989, Prog. Math. 85, 349-366 (1990).
Let \(\Phi_n(z)=\sum^{\varphi(n)}_{m=0} a(m,n) z^n\) be the \(n\)-th cyclotomic polynomial and let \(S(n)=\sum^{\varphi (n)}_{m=0} \vert a(m,n)\vert\). The author shows that if \(\varepsilon(n)\) is a function defined for all positive integers such that \(\varepsilon(n)\to 0\) as \(n\to \infty\), then \(S(n)\ge n^{1+\varepsilon(n)}\) for almost all \(n\). This settles a conjecture of Erdős that, for any constant \(c>0\), we have \(\max_{0\leq m\leq \varphi(n)} \vert a(m,n)\vert \geq c\) for almost all \(n\).
[For the entire collection see Zbl 0711.00008.]

11C08 Polynomials in number theory
11B83 Special sequences and polynomials