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

Abstract

Let $\alpha$ be a nonzero algebraic integer of degree $d$, all of whose conjugates ${\alpha }_1=\alpha ,{\alpha }_2,\dots ,{\alpha }_d$ lie in a sector $|argz|\le 𝜃$, $0<𝜃\le 90^{\circ }$. We define the S-measure of $\alpha$ by $S\left(\alpha \right)={\sum }_{i=1}^d\left|{\alpha }_i\right|$ and the absolute $S$-measure of $\alpha$ by $\text{ s}\left(\alpha \right)=S\left(\alpha \right)∕d$. We compute the greatest lower bound $c\left(𝜃\right)$ of $\text{ s}\left(\alpha \right)$ for $\alpha$ belonging to twelve subintervals of $\left(0,𝜃\right)$. Among these subintervals, three are complete. These computations use the principle of explicit auxiliary functions and our recursive algorithm.