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Anatomy of integers and cyclotomic polynomials. (English) Zbl 1186.11010
De Koninck, Jean-Marie (ed.) et al., Anatomy of integers. Based on the CRM workshop, Montreal, Canada, March 13–17, 2006. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4406-9/pbk). CRM Proceedings and Lecture Notes 46, 89-95 (2008).

Let $n={p}_{1}{p}_{2}\cdots {p}_{\omega \left(n\right)}$ with primes ${p}_{1}>{p}_{2}>\cdots >{p}_{\omega \left(n\right)}>2$, and let ${{\Phi }}_{n}$ be the $n$-th cyclotomic polynomial. The author shows that for $|z|=1$:

$log|{{\Phi }}_{n}\left(z\right)|\le \sum _{k=1}^{\omega \left(n\right)}{2}^{k}log\left({p}_{k}\right)+O\left({2}^{\omega \left(n\right)}\right)·$

##### MSC:
 11C08 Polynomials (number theory) 11N56 Rate of growth of arithmetic functions
##### Keywords:
cyclotomic polynomials