On binary cyclotomic polynomials

We study the number of nonzero coefficients of cyclotomic polynomials Φm where m is the product of two distinct primes. 1. Presentation of the results Let m > 1 be an integer and Φm the cyclotomic polynomial defined by the equality Φm(X) := m ∏ j=1 (j,m)=1 ( X − exp(2πij/m) ) . This monic polynomial belongs to Z[X] and its degree is equal to φ(m), the Euler function of the integer m. Let θ(m) be the number of nonzero coefficients of Φm. Of course, θ(m) satisfies the trivial inequalities 2 ≤ θ(m) ≤ φ(m) + 1, which are optimal when one considers the cases m = 1 or m = p a prime number. In these cases, all the coefficients of Φm are equal to 1. We make the convention to reserve the letters p and q to prime numbers and we say that the integer m is binary if it is of the form m = pq, with distinct p and q. Let B = {6, 10, 14, 15, 21, . . . } be the set of binary integers. For m ∈ B, we say that the associated cyclotomic polynomial Φm is binary. The coefficients of the binary cyclotomic polynomial Φm are equal to 0, 1 or −1. Furthermore, in that particular case, the function θ(m) has an explicit expression in terms of p and q which can be exploited by analytic number theory (see Proposition 1 below). Recently Bzdȩga [4] started the study of the distribution function of the map m ∈ B 7→ θ(m). To recall his results, let γ and x satisfying 0 0, there exists C(γ), c( , γ) > 0 and x0 = x0( , γ), such that, for x ≥ x0 one has the inequalities (2) c( , γ)x 1 2 +γ− ≤ Hγ(x) ≤ C(γ)x 1 2 +γ . Date: February 15, 2012. 2010 Mathematics Subject Classification. Primary 11N13 ; Secondary 11L20 .