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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.]

MSC:
 11C08 Polynomials in number theory 11B83 Special sequences and polynomials
Keywords:
coefficients; cyclotomic polynomial