  Professor Georges Rhin considers a nonzero algebraic integer $\a$ with
conjugates $\a_1=\a, \ldots, \a_d$ and asks what can be said about $\d \sum_{ |
\a_i | >1} | \a_i |$, that we denote ${\rm{R}}(\a)$. If $\a$ is supposed to be
a totally positive algebraic integer, we can establish an analog to the famous
Schur-Siegel-Smyth trace problem for this measure. After that, we compute the
greatest lower bound $c(\theta)$ of the quantities ${\rm{R(\a)}}/d$, for $\a$
belonging to nine subintervals of $]0, 90 [$. The third three subintervals are
complete and consecutive. All our results are obtained by using the method of
explicit auxiliary functions. The polynomials involved in these functions are
found by our recursive algorithm.
