The integer Chebyshev constant of Farey intervals

We obtain new bounds for the integer Chebyshev constant of intervals $[p/q,r/s]$ where $p$, $q$, $r$ and $s$ are non-negative integers such that $q\,r-p\,s=1$. As a consequence of the methods used, we improve the known lower bound for the trace of totally positive algebraic integers.