Zentralblatt MATH

Publications of (and about) Paul Erdös
Zbl.No:  081.01703
Autor:  Erdös, Pál
Title:  On the growth of the cyclotomic polynomial in the interval (0,1). (In English)
Source:  Proc. Glasg. Math. Assoc. 3, 102-104 (1957).
Review:  Suppose n is a positive integer greater than unity and Fn(x) is the n-th cyclotomic polynomial. Let An be the largest absolute value of any coefficient of Fn(x), let Bn be the maximum value taken on by Fn(x) on the interval [0,1], and let Cn be the maximum value taken on by Fn(x) on the disc |x| \leq 1. In a previous paper (Zbl 038.01004) the author has shown that there is a positive constant c such that

Cn > \exp\exp{c log n/ log log n}

for infinitely many values of n. Since An < Cn \leq nAn, this is equivalent to the corresponding assertion for An.
In the present paper the author gives a simpler proof of the more specific assertion that

Bn > \exp\exp{c log n log log n}    (*)

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 Fn(x) at a carefully chosen value of x slightly less than 1-n- ½. (Since Fn(0) = Fn(1) = 1 if n has more than one prime factor, the maximum value of Fn(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 1/4 log 2, and a slight modification of the argument will give (^*) for any c less than 2/7 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 035.31102), it is almost immediate that if \epsilon > 0, then

Bn \leq Cn \leq nAn < \exp\exp{(1+\epsilon)(log 2) log n/ log log n}

for all large n.
Reviewer:  P.T.Bateman
Classif.:  * 11C08 Polynomials
Index Words:  linear algebra, polynomials, forms

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