Cyclotomic polynomials with many primes dividing their orders

Abstract

The maximum of the absolute value of the n-th cyclotomic polynomial Φn is estimated from below in terms of the number ω(n) of prime divisors. If ω(n )i s abnormally large ( ≥ C log log n with suitable C) then a lower bound for the maximum A(n) of the absolute values of the coefficients of Φn follows. The result generalizes a result of the second author, see [7], in that it no longer allows exceptions. Also the proof is simpler. Moreover a rather small bound for the third logarithmic momentum of |Φn(z)| on the unit circle is obtained.