By analogy with the trace of an algebraic integer $\alpha $ with conjugates $\alpha _1=\alpha , \ldots , \alpha _d$ , we define the G-measure $ {\mathrm {G}} (\alpha )= \sum _{i=1}^d ( |\alpha _i| + 1/ | \alpha _i | )$ and the absolute ${\mathrm G}$ -measure ${\mathrm {g}}(\alpha )={\mathrm {G}}(\alpha )/d$ . We establish an analogue of the Schur–Siegel–Smyth trace problem for totally positive algebraic integers. Then we consider the case where $\alpha $ has all its conjugates in a sector $| \arg z | \leq \theta $ , $0 < \theta < 90^{\circ }$ . We compute the greatest lower bound $c(\theta )$ of the absolute G-measure of $\alpha $ , for $\alpha $ belonging to $11$ consecutive subintervals of $]0, 90 [$ . This phenomenon appears here for the first time, conforming to a conjecture of Rhin and Smyth on the nature of the function $c(\theta )$ . All computations are done by the method of explicit auxiliary functions.

Let $\alpha $ be a totally positive algebraic integer of degree d, with conjugates $\alpha _1=\alpha , \alpha _2, \ldots , \alpha _d$ . The absolute $S_k$ -measure of $\alpha $ is defined by $s_k(\alpha )= d^{-1} \sum _{i=1}^{d}\alpha _i^k$ . We compute the lower bounds $\upsilon _k$ of $s_k(\alpha )$ for each integer in the range $2\leq k \leq 15$ and give a conjecture on the results for integers $k>15$ . Then we derive the lower bounds of $s_k(\alpha )$ for all real numbers $k>2$ . Our computation is based on an improvement in the application of the LLL algorithm and analysis of the polynomials in the explicit auxiliary functions.

Let α be a totally positive algebraic integer of degree d≥2 and α1=α,α2,…,αd be all its conjugates. We use explicit auxiliary functions to improve the known lower bounds of Sk/d, where Sk=∑ di=1αki and k=1,2,3. These improvements have consequences for the search of Salem numbers with negative traces.