On the growth of the cyclotomic polynomial in the interval (0,1).

*(English)*Zbl 0081.01703Suppose \(n\) 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 \(n\).

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 |