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On the growth of the cyclotomic polynomial in the interval (0,1). (English) Zbl 0081.01703
Suppose $$n$$ is a positive integer greater than unity and $$F_n(x)$$ is the $$n$$-th cyclotomic polynomial. Let $$A_n$$ be the largest absolute value of any coefficient of $$F_n(x)$$, let $$B_n$$ be the maximum value taken on by $$F_n(x)$$ on the interval $$[0,1]$$, and let $$C_n$$ be the maximum value taken on by $$F_n(x)$$ on the disc $$|x| \leq 1$$. In a previous paper (Zbl 0038.01004) the author has shown that there is a positive constant $$c$$ such that $C_n > \exp\exp\{c\log n/\log\log n\}$ for infinitely many values of $$n$$. Since $$A_n < C_n \leq nA_n$$, this is equivalent to the corresponding assertion for $$A_n$$.
In the present paper the author gives a simpler proof of the more specific assertion that $B_n > \exp\exp\{c \log n\log\log n\}\tag{*}$ for infinitely many values of $$n$$, where $$c$$ is a suitably chosen positive number. The values of $$n$$ considered are products of a large number of very nearly equal primes and for these values of $$n$$ the author investigates $$F_n(x)$$ at a carefully chosen value of $$x$$ slightly less than $$1-n^{-1/2}$$. (Since $$F_n(0) = F_n(1)=1$$ if $$n$$ has more than one prime factor, the maximum value of $$F_n(x)$$ on $$[0,1]$$ occurs at an interior point of the interval.) The argument requires only elementary results on the distribution of prime numbers. Although the author does not calculate $$c$$ explicitly, his proof will give $$(^*)$$ for any $$c$$ less than $${1 \over 4}\log 2$$, and a slight modification of the argument will give $$(^*)$$ for any $$c$$ less than $${2 \over 7}\log 2$$. The author believes that perhaps $$(^*)$$ holds for any $$c$$ less than $$\log 2$$, but that the present method of proof is not strong enough to give such a result. On the other hand, this would be as far as one could go, since, as the reviewer has remarked (cf. Zbl 0035.31102), it is almost immediate that if $$\varepsilon >0$$, then $B_n \leq C_n \leq nA_n < \exp\exp\{(1+\varepsilon)(\log 2)\log n/\log\log n\}$ for all large $$n$$.
Reviewer: P.T.Bateman

MSC:
 11C08 Polynomials in number theory
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