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

Abstract

Let α be a nonzero algebraic integer of degree d, all of whose conjugates α1=α,α2,…,αd lie in a sector |argz|≤𝜃, 0<𝜃≤90∘. We define the S-measure of α by S(α)= ∑i=1d|αi| and the absolute S-measure of α by s(α)=S(α)∕d. We compute the greatest lower bound c(𝜃) of s(α) for α belonging to twelve subintervals of (0,𝜃). Among these subintervals, three are complete. These computations use the principle of explicit auxiliary functions and our recursive algorithm.