On the size of the coefficients of the cyclotomic polynomial. (English) Zbl 0512.10005

Bateman, Paul T.

Sémin. Théor. Nombres, Univ. Bordeaux I 1981-1982, Exp. No. 28, 17 p. (1982).

Write the \(n\)-th cyclotomic polynomial as \(\prod (1 - \rho z) = \sum a(m,n)z^n\), the product being over all primitive \(n\)-th roots, \(\rho\), of unity. Define \(A(n) = \max \vert a(m,n)\vert\). The author gives a previously unpublished proof, due to B. Saffari, of a result of R. C. Vaughan in [Mich. Math. J. 21, 289–295 (1975; Zbl 0304.10008)] which is a sharpening of the assertion that \[ \log \log A(n) > \log 2 \log n/\log \log n \quad\text{for infinitely many }n. \]
Other results of the paper are taken from a forthcoming article of the author with R. C. Vaughan and C. Pomerance in [Topics in classical number theory, Colloq. Budapest 1981, Vol. I, Colloq. Math. Soc. János Bolyai 34, 171–202 (1984; Zbl 0547.10010)]. For example, it is shown that \[ A(n)\le n^{f(k)} : f(k)= (2^{k-1}/k) - 1 \] for all \(n\) having exactly \(k\) distinct odd prime factors. They conjecture that this is best possible in the sense that \(A(n)\ge c_kn^{f(k)}\) for infinitely many such \(n\), but what they prove in this direction is somewhat weaker.

Reviewer: George Greaves (Cardiff)

MSC:

11B83	Special sequences and polynomials
11C08	Polynomials in number theory

Keywords:

size of coefficients; cyclotomic polynomial

Citations:

Zbl 0304.10008; Zbl 0547.10010