The N-measure for algebraic integers having all their conjugates in a sector

International audienceLet α be a nonzero algebraic integer of degree d whose all conjugates α 1 = α, α 2 ,. .. , α d lie in a sector | arg z| ≤ θ, 0 ≤ θ ≤ 90 •. We define the N-measure of α by N(α) = d i=1 (|α i |+1/|α i |) and the absolute N-measure of α by ν(α) = N(α) 1/ deg(α). Firstly, we consider the case θ = 0. We prove that N(α) ∈ N and that, if α is a reciprocal algebraic integer, N(α) is a square. Then, we study the set N of the quantities ν(α). We prove that there exists a number l such that N is dense in (l, ∞). Finally, using the method of auxiliary functions, we find the seven smallest points of N in (2, l). In case of 0 < θ ≤ 90 • , we compute the greatest lower bound c(θ) of the absolute N-measure of α, for α belonging to eight subintervals of ]0, θ[. Among these subintervals, two are complete. These computations are also done by using auxiliary functions. The polynomials involved in the auxiliary functions are found by our recursive algorithm