Cyclotomic polynomials with many primes dividing their orders
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CYCLOTOMIC POLYNOMIALS WITH MANY PRIMES DIVIDING THEIR ORDERS Sergei Konyagin∗ (Moscow) Helmut Maier (Ulm) and Eduard Wirsing (Ulm) [Communicated by: Andr´ as S´ ark¨ ozy ]
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) is 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.
1. Some notation Φn (x) = m a(m, n)xm is the n-th cyclotomic polynomial, ω(n) the number of prime divisors of n, Z∗ an abbreviation for Z {0}, e(α) = e2πiα . Underscoring denotes triplets: d = (d1 , d2 , d3 ). If, however, (d1 , . . . , dk ) denotes an integer then it is the greatest common divisor. A logic expression A in {. . .} means its characteristic function: {A} = 1 if A applies and 0 otherwise.
Mathematics subject classification number: 11N56. Key words and phrases: cyclotomic polynomials, maximum of coefficients, number of prime divisors, logarithmic momentums. * During the time of the work the first author was supported by grants 02-01-00248 from the Russian Foundation for Basic Regarch and RFN NSh-3004-2003.1. 0031-5303/2004/$20.00 c Akad´ emiai Kiad´ o, Budapest
Akad´ emiai Kiad´ o, Budapest Kluwer Academic Publishers, Dordrecht
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s. konyagin, h. maier and e. wirsing
2. Introduction The coefficients a(m, n) of the cyclotomic polynomials and in particular their maximum A(n) := maxm |a(m, n)| have been the subject of numerous investigations (see P. T. Bateman, C. Pomerance, R. C. Vaughan [1] and the references given there). If ε and ψ are any functions on N such that ε(n) → 0, ψ(n) → ∞ as n tends to infinity, then one of us, see [4, 5, 6], has shown that nε(n) ≤ A(n) ≤ nψ(n)
for almost all integers n
and that these inequalities are best possible. For almost all integers n, as is wellknown, one has |ω(n) − log log n| ≤ (log log n)1/2+ε ,
ε > 0, arbitrary.
In [7] the same author considered numbers n for which ω(n) exceeds this range, more precisely the numbers in the set E(C, x) := {n ≤ x : ω(n) ≥ C log log n, n square-free}, where C is any constant > 2/ log 2, and showed that log 2 A(n) ≥ exp (log n)C 2 −ε
(1)
holds for almost all of them. The proof rests on a comparison of the second and the third of the momentums 1 j Mj (n) := log Φn e(α) dα . 0
The second can be determined explicitely, M2 (n) =
π 2 ϕ(n) ω(n) 2 , 12n
(2)
whereas in [7] it could be shown that |M3 (n)| ≤ 2ω(n) (log n)ε
for most n ∈ E(C, x) .
(3)
In the present paper we prove an estimate like (3), but for all n without exception, and in fact with a smaller bound. Correspondingly, the estimate for A(n) can now be given for all n with the specified large number of prime divisors.
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coefficients
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