## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  295.10014
Autor:  Erdös, Paul; Vaughan, R.C.
Title:  Bounds for the r-th coefficients of cyclotomic polynomials. (In English)
Source:  J. London Math. Soc., II. Ser. 8, 393-400 (1974).
Review:  Let \Phin(z) = sum \phi (n)r = 0ar(n)zr be the n-th cyclotomic polynomial and ar(n): = 0 for r > \phi (n), where \phi is Euler's function. The following estimates are proved:

|\Phin(z)| < \exp (\tau (1- |z|)-1+C1(1- |z|)-3/4)

with \tau: = prodp in P (1-{2 \over p(p+1) ) for each z in C with |z| < 1, |ar(n)| < \exp (2 \tau ½r ½+C2r3/8), maxn in N |ar(n)| > \exp (C3r ½ log ^{- ½r) for r > r0. The second result follows immediately from the first. The third estimate requires three lemmas on the representation of numbers as sums of primes.
Reviewer:  H.Möller
Classif.:  * 11B39 Special numbers, etc.

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